---
title: "Time Series Homework: Chapter 4 Lesson 1"
subtitle: "Please_put_your_name_here"
format: 
  html:
    embed-resources: true
    toc: true
---

```{r, echo=FALSE, message=FALSE, warning=FALSE}

# Loading R packages
if (!require("pacman")) install.packages("pacman")
pacman::p_load(tidyverse, 
               tsibble, fable,
               feasts, tsibbledata,
               fable.prophet,
               patchwork,
               lubridate, 
               rio,
               ggplot2,
               kableExtra
               )
```

## Questions

### Question 1 - Key Definitions (10 points)

Answer the prompt to the learning outcome below. Include any mathematical expressions or illustrations that may accompany the definitions and ideas if available.

::: {.callout-note title="Answer" icon="false"}
-   **Define Gaussian white noise** <!-- Delete this and put your answer here. -->

-   **State DWN second order properties** <!-- Delete this and put your answer here. -->

-   **State the assumptions needed to categorize residual error series as white noise** <!-- Delete this and put your answer here. -->

-   **Define a random walk** <!-- Delete this and put your answer here. -->
:::

### Question 2 - Gaussian White Noise (25 points)

#### a)  Simulate a realization of Gaussian White Noise and plot it. Use 25 points and $\sigma^2=1$

::: {.callout-note title="Answer" icon="false"}
```{r}



```
:::

#### b)  Estimate the second order properties of the sample. Why don't they match exactly with the parameters you set to create it?

::: {.callout-note title="Answer" icon="false"}
```{r}



```

<!-- Delete this and put your answer here. The three colons mark the end of your answer. -->
:::

#### c)  Simulate a realization of Gaussian White Noise and plot it. Use 500 points and $\sigma^2=1$

::: {.callout-note title="Answer" icon="false"}
```{r}



```
:::

#### d)  Estimate the second order properties of the sample in part c. Contrast your results with part b? Please explain your results.

::: {.callout-note title="Answer" icon="false"}
```{r}



```

<!-- Delete this and put your answer here. The three colons mark the end of your answer. -->
:::

### Question 3 - Random Walks (45 points)

#### a)  Simulate a realization of a white noise process using the Exponential distribution. If $s_t$ is the realization from an exponentially distributed random variable, use the white noise $w_t=s_t-\lambda$ for your simulation. The length of the series is 500 points and $\lambda=1$. Please plot your simulation.

::: {.callout-note title="Answer" icon="false"}
```{r}



```
:::

#### b)  Superimpose a histogram of the Gaussian White noise simulation from Question 2 versus the Exponential White Noise. Please compare and contrast the two distributions. Your chart should look close to Figure 1 below. *Hint: research exponential random variables and their right-tail properties.*

    ![Figure 1](https://github.com/byuistats/timeseries/raw/master/images/Overlayed_Histograms_HW4_1.png)

::: {.callout-note title="Answer" icon="false"}
```{r}



```
:::

#### c)  Suppose you were to create two random walk series using the Gaussian and Exponential DWN simulations. How do you think they would differ from one another?

::: {.callout-note title="Answer" icon="false"}
<!-- Delete this and put your answer here. The three colons mark the end of your answer. -->
:::

#### d)  Create two random walk series using the Gaussian and Exponential DWN simulations. Plot them together. Are there some conclusions you can draw from comparing the series? *Hint: you might want to generate multiple samples before you start making observations.*

::: {.callout-note title="Answer" icon="false"}
```{r}



```

<!-- Delete this and put your answer here. The three colons mark the end of your answer. -->
:::

### Rubric

+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| **Criteria**                                         |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                      | **Mastery (10)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | **Incomplete (0)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| #### **Question 1: Definitions**                     | The student correctly defined each of the terms and included mathematical expressions or illustration if available in the text or the Time Series Notebook                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | The student did not provide a correct definition for one or more of the terms.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                      | **Mastery (5)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | **Incomplete (0)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| #### **Question 2a: Gaussian White Noise**           | They effectively plot the simulated realization of Gaussian White Noise in R using plotting functions like plot() or ggplot2, ensuring proper labeling of the axes, a title, and any other necessary elements for clear visualization. Code is well-commented.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | Students struggle to simulate Gaussian White Noise in R or effectively plot the simulation, potentially due to errors in coding or misunderstandings of the process. Their R code may lack sufficient comments, making it challenging for others to understand or reproduce the results. The plot may lack clarity or proper presentation, hindering interpretation, and may omit necessary elements such as axis labels or a title.                                                                                                                                                                                                                                                                                                                         |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                      | **Mastery (10)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | **Incomplete (0)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| #### **Question 2b: Sample second order properties** | Students demonstrate an understanding of statistical concepts such as variance, covariance, and autocorrelation, and use appropriate methods or functions to estimate these properties in their chosen statistical software. Proficient explanations may highlight factors such as sample size, random variability, or measurement error, which can introduce discrepancies between estimated properties and the parameters set during data creation.                                                                                                                                                                                                                                                                                                    | Students fail to estimate the second-order properties of the sample data or provide a coherent explanation for discrepancies with the parameters set during data creation. They may demonstrate a limited understanding of statistical concepts or use inappropriate methods for estimation. Additionally, they may overlook key factors contributing to discrepancies, providing vague or incomplete explanations. Overall, their analysis may lack depth or clarity, hindering comprehension of the reasons behind the observed differences.                                                                                                                                                                                                               |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                      | **Mastery (5)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | **Incomplete (0)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| #### **Question 2c: Gaussian White Noise**           | They effectively plot the simulated realization of Gaussian White Noise in R using plotting functions like plot() or ggplot2, ensuring proper labeling of the axes, a title, and any other necessary elements for clear visualization. Code is well-commented.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | Students struggle to simulate Gaussian White Noise in R or effectively plot the simulation, potentially due to errors in coding or misunderstandings of the process. Their R code may lack sufficient comments, making it challenging for others to understand or reproduce the results. The plot may lack clarity or proper presentation, hindering interpretation, and may omit necessary elements such as axis labels or a title.                                                                                                                                                                                                                                                                                                                         |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                      | **Mastery (10)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | **Incomplete (0)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| #### **Question 2d: Sample second order properties** | Responses accurately estimate the second-order properties of the sample data in part c and effectively contrast these results with those from part b, which had a smaller sample size. Students demonstrate an understanding of how larger sample sizes in part c may lead to more reliable estimates due to reduced sampling variability.                                                                                                                                                                                                                                                                                                                                                                                                               | Responses below expectations may struggle to accurately estimate the second-order properties of the sample data in part c or provide a coherent contrast with part b. Additionally, they may provide vague or superficial explanations for the observed discrepancies, lacking depth or clarity in their analysis. Overall, their response may lack insight into the statistical properties of sample estimators and their implications for data analysis or interpretation.                                                                                                                                                                                                                                                                                 |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                      | **Mastery (10)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | **Incomplete (0)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| #### **Question 3a: Simulation and Plot**            | Responses simulate a realization of a white noise process using the Exponential distribution as instructed. The resulting plot effectively visualizes the simulation, showcasing the characteristics of the white noise process. Proficient submissions ensure clarity in the plot, with proper labeling of axes and a title, facilitating interpretation of the simulated white noise.                                                                                                                                                                                                                                                                                                                                                                  | Authors may struggle to accurately simulate a realization of a white noise process using the Exponential distribution or generate the corresponding white noise series. They may encounter difficulties in implementing the instructions or understanding the relationship between the Exponential distribution and white noise. The resulting plot may lack clarity or proper presentation, hindering interpretation of the simulated white noise. Additionally, they may omit necessary elements such as axis labels or a title, further detracting from the clarity of the visualization.                                                                                                                                                                 |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                      | **Mastery (10)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | **Incomplete (0)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| #### **Question 3b: Histograms**                     | Responses superimpose a histogram of the Gaussian White Noise simulation from Question 2 alongside the Exponential White Noise. They compare and contrast the two distributions, demonstrating an understanding of statistical concepts such as mean and variance. Proficient analyses may include research findings on the mean and variance of an exponential random variable, providing context for the comparison. The resulting chart closely resembles Figure 1, showing clear distinctions between the distributions and offering insightful commentary on their similarities and differences.\                                                                                                                                                   | Responses may struggle to accurately superimpose the histograms of Gaussian White Noise and Exponential White Noise or provide a coherent comparison between the two distributions. They may lack understanding of statistical concepts such as mean and variance or fail to research the properties of an exponential random variable. The resulting chart may lack clarity or proper presentation, hindering interpretation of the distributions. Additionally, they may overlook key differences between the distributions or provide vague or superficial comparisons, indicating a limited understanding of the underlying concepts.                                                                                                                    |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                      | **Mastery (10)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | **Incomplete (0)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| #### **Question 3c: Educated guessing**              | Responses offer insightful analysis of the differences between two random walk series created using Gaussian and Exponential Distributed White Noise (DWN) simulations. They demonstrate an understanding of random walk processes and how they are influenced by the characteristics of the underlying noise. Proficient analyses may discuss how the variance of the noise affects the volatility of the random walk series, noting that Gaussian DWN typically results in series with constant variance while Exponential DWN may lead to series with time-varying volatility. Additionally, they may consider the long-term behavior of the random walk series and how it is influenced by the properties of the underlying noise distribution.      | Students provide limited or superficial analysis of the differences between random walk series created using Gaussian and Exponential DWN simulations. They may demonstrate a lack of understanding of random walk processes or fail to consider how the characteristics of the noise distribution impact the behavior of the series. Additionally, they may overlook key differences between Gaussian and Exponential DWN or provide vague or unsupported explanations. Overall, their analysis may lack depth or coherence, indicating a limited understanding of the concepts involved.                                                                                                                                                                   |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                      | **Mastery (20)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | **Incomplete (0)**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| #### **Question 4d: Random Walks**                   | Students create two random walk series using Gaussian and Exponential Distributed White Noise (DWN) simulations and plot them together. Their analysis draws clear, insightful conclusions from comparing these series, focusing on aspects such as the differences in volatility, the distribution of increments, and the overall trend behavior. They might note the constant versus variable variance in the series and how the underlying DWN simulations influence the random walks' paths. Their plots accurately represent the series, and their commentary connects observed differences to theoretical expectations, demonstrating a strong grasp of random walk characteristics and the implications of using Gaussian versus Exponential DWN. | Author struggles to create or plot the random walk series using Gaussian and Exponential DWN simulations. The comparisons of the series may be vague, incorrect, or superficial, with limited analysis of the differences in volatility, trends, or distribution of increments. They may fail to connect observed behaviors of the series to the properties of Gaussian versus Exponential DWN, demonstrating a lack of understanding of the underlying statistical concepts. The plots may be poorly constructed, making it difficult to observe key characteristics or compare the series effectively. Overall, their analysis lacks depth, coherence, or accuracy, indicating a weak grasp of how different DWN simulations influence random walk series. |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| **Total Points**                                     | **90**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
