Assignments

---

Homework08: (complete by Monday December 2)

For the math problems, scan your handwritten work to one .pdf file.

• Read: Ch. 3

• Exercises Ch. 3: 3.25 comp, 3.27 comp, 3.28 comp

Additional Exercise 1: Submit your .html and .qmd files.

Using the code in the Handout: Likelihood Ratio Test examples

• Likelihood Ratio Test examples
  • likelihood_ratio.html
  • likelihood_ratio.qmd

Change the Reduced Model to include be the model with ‘hp’ and ‘cyl’. And for the Logistic Regression change the model to include ‘spontaneous’.

Determine the results of the tests.

---

Homework07: (complete by Monday November 18)

For the math problems, scan your handwritten work to one .pdf file.

• Read: Ch. 3, Ch. 5

• Exercises Ch. 3: 3.5 math, 3.12 math, 3.14 math, 3.16 math, 3.18 comp (Use Python or R) SOCELL.csv

Additional Exercise 1: Submit your .html and .qmd files.

Run the following R code in the following R Quarto Notebook to simulate data and use the fitdistr function to estimate the parameters in the models from handout1a.pdf.

• ft_mle.html
• ft_mle.qmd

1. Change the parameters used in each of the simulations and refit the models.

   1. Change theta <- 0.5 to a different value.
   2. Change theta <- 5 to a different value.
   3. Change theta <- 5 to a different value.
   4. Change theta <- 0.5 to a different value.

Additional Exercise 2: Complete the following problem in an R Quarto Notebook. Submit your .html and .qmd files.

• prob2_suess.pdf
• whale.txt

Note: The solution to the Additional Exercise 2 has been posted in Canvas.

Note: The R package for the book fell off of CRAN April 2024. To install the package in R you can download the last version of the mistat archive and Install From: Package Archive File .tar.gz

---

Homework06: (complete by Monday November 4)

For the math problems, scan your handwritten work to one .pdf file. For the code problems, if you are using Colab, submit a .docx file containing the shared link to your Colab Notebook.

• Read: Ch. 3, Ch. 5
• Exercises Ch. 3: 3.2 math, 3.10 math, 3.11 math, 3.13 math
• Complete the exercises related to the M.L.E. in this handout. handout1a.pdf

---

Homework05: (complete by Monday October 14)

For the math problems, scan your handwritten work to one .pdf file. For the code problems, if you are using Colab, submit a .docx file containing the shared link to your Colab Notebook.

• Read: Ch. 2

• Exercises Ch. 2: 2.67 math, 2.68 math, 2.69 math, 2.71 math, 2.73 math, 2.101 math, 2.102 math, 2.103 math, 2.104 math, 2.105 comp

• Read the Mean and Standard Deviation Simulation Stats Magazine paper. Mean&SDSimulation.pdf

• Complete the following Exercises using Python.

Additional Exercise 1: (L.L.N.) Demonstrate the Law of Large Numbers for the Normal Distribution.

• Sample \(n = 1000\) Standard Normal random values and put them in a vector \(z\).
• Write an Python function to compute the cumulative mean.
• Create a vector \(x = (1, 2, ..., n)\).
• Compute the cumulative mean vector \(y\) from \(z\) and \(x\).
• Approximate the probability that \(|y| > \epsilon\).

Additional Exercise 2: (C.L.T.) Demonstrate the Central Limit Theorem for the Normal and Exponential Distributions.

• Sample \(k = 1000\) samples of size \(n = 30\) from the \(Normal(\mu = 5, \sigma^2 = 2^2 = 4)\).
• Compute the means for each of the \(k\) samples.
• Draw a histogram of these \(k\) sample means.
• Describe the shape of the histogram.
• Do the above steps again with the \(Exponential(\lambda = 1/3)\).

Additional Exercise 3: Demonstrate the independence of the sample mean and sample variance when sampling from the \(Normal(\mu = 5, \sigma^2 = 2^2 = 4)\).

• Sample \(k = 1000\) samples of size \(n = 30\) from the \(Normal(\mu = 5, \sigma^2 = 2^2 = 4)\).
• Compute the sample means and standard deviations for each of the \(k\) samples.
• Plot the sample means versus the sample standard deviations.
• Compute the correlation between the sample means and standard deviations.
• Do the above steps again with the \(Exponential(\lambda = 1/3)\).

---

Homework04: (complete by Monday September 23)

For the math problems, scan your handwritten work to one .pdf file. For the code problems, if you are using Colab, submit a .docx file containing the shared link to your Colab Notebook.

• Read: Ch. 2
• Exercises Ch. 2: 2.24 math, 2.60 math, 2.62 math, 2.63 comp, 2.76 math

---

Homework03: (complete by Monday September 16)

For the math problems, scan your handwritten work to one .pdf file. For the code problems, if you are using Colab, submit a .docx file containing the shared link to your Colab Notebook.

• Read: Ch. 2
• Read: Birthday Problem Stats Magazine Paper. Birthday.pdf
• Exercises Ch. 2: 2.20 math, 2.21 math, 2.27 math and code, 2.29 math, 2.41 math, 2.42 code

Hint: For 2.27, assume that a stick being randomly broken at two spots is equivalent to randomly selecting two points on a stick. Assume the length of the stick is 1. Assume that the two points are uniformly distributed on the stick. What is the probability that the three pieces can form a triangle?

Try using Colab Generate to produce Python code to simulate the probability of the three pieces forming a triangle. You can use the following code Prompts to get started.

Prompts

1. prompt: Using Python code simulate data from a bivariate distribution where the input variable has a Poisson distribution with mean 100 and the distribution of X given N is Binomial with p = 0.95. Plot a scatterplot of the simulated data.

2. prompt: Fit a linear regression model to the data.

3. prompt: Fit the same model without an intercept term.

4. prompt: Compute the mean and standard deviation of the X data.

5. prompt: Make prediction of X given N = 75 and N = 120.

---

Homework02: (complete by Wednesday September 4)

For the math problems, scan your handwritten work to one .pdf file. For the code problems, if you are using Colab, submit a .docx file containing the shared link to your Colab Notebook.

• Read: Ch. 2
• Exercises Ch. 2: 2.1 math, 2.2 math, 2.8 math, 2.10 math and code, 2.13 math and code, 2.14 math and code
• Read: the Handouts on random number generation.
  • From jStor find and read the following papers. JSTORE - online statistics and probability articles, available on campus. See CSUEB Libary > Databases A-Z > J
  • Leemis, Lawrence, Relationships Among Common Univariate Distributions, The American Statistician, Vol. 40, No. 2 (May 1986), pp. 143-146.
• Complete the following Exercises using Python.

Additional Exercise 1: Using Python verify that these models are the same: GAMMA(\(\alpha = n/2, \beta = 1/2\)) = CHISQ(\(n\)). To show this compute the cumulative probabilities for the two models using the same value of \(x\) and parameter \(n\).

---

Homework01: (complete by Monday Aug. 26)

For the math problems, scan your handwritten work to one .pdf file. For the code problems, if you are using Colab, submit a .docx file containing the shared link to your Colab Notebook.

• Read: Preface
• Read: Appendix A, Appendix B
• Read: Ch. 1
• Exercises Ch. 1: 1.3 code, 1.4 code, 1.8 math, 1.9 math, 1.22 code
• Example Solution: Stat640 Homework01.ipynb

---