MATH 314 Homework 06

Due 2025-10-13 by 11:59pm

1. Type, using inside your jupyter lab notebook, your solution to the likelihood problem for an Exponential distribution.

2. Pick a member of the Rayleigh family of distributions by choosing a scale parameter .

   a. Generate Rayleigh random variables based on your choice of . Store these data as a variable named x.

   b. Write a log-likelihood function in Python that takes two arguments: theta and data. Don't forget that most scientific software has functions for minimization, not maximization.

   c. Call the scipy.optimize function minimize on your log-likelihood function from b. above, to find the maximum likelihood estimate of your choice of based on the data you generated in a.

   d. Make a plot of the log-likelihood function with the data generated above as fixed.

3. Pick a member of the Normal family of distributions by choosing a location parameter and a scale parameter .

   a. Generate Normal random variables based on your choice of . Store these data as a variable named x.

   b. Write a log-likelihood function in Python that takes two arguments: theta and data. Don't forget that most scientific software has functions for minimization, not maximization.

   c. Call the scipy.optimize function minimize on your log-likelihood function from b. above, to find the maximum likelihood estimate of your choice of based on the data you generated in a. Don't forget that you're now estimating two parameters, so the initial values of is specified as a numpy array of two numbers.