Assignments
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
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.
prompt: Fit a linear regression model to the data.
prompt: Fit the same model without an intercept term.
prompt: Compute the mean and standard deviation of the X data.
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