Prof. Vanja M. Dukic  

Appm 7400-001    Spring 2012
http://amath.colorado.edu/courses/7400_001/2012Spr

Bayesian Statistics and Computing

• Problem 1. Suppose that you have 10 components A and 5 components B in a particular lab machine. Assume that the lifetimes of all components are independent and exponentially distributed. Furthermore, assume that the mean lifetime of each component A w hours, and that the lifetime of each component B is 2w hours. You turn the machine on, and leave it operating alone for 24 hours. Upon return you notice that 5 of 10 components A have failed, and 1 component B.
  
  a. What is the exact probability that 5 components A and 1 component B fail within 24 hours?
  
  b. What are the likelihood and log-likelihood functions of w?
  
  c. What value of w maximizes the likelihood function?





• Problem 2. A company that makes hot tubs wants to claims that 99% of the time their tubs take no longer than x minutes to reach the 100F temperature. During testing, the following data on time-to-100F (in minutes) were recorded for 20 randomly selected hot tubs:
  
  16.26 18.63 14.09 15.37 19.80 11.86 15.29 17.34 15.53 15.70 13.38 12.62 13.17 13.39 16.75 13.07 13.70 14.04 11.33 17.31
  
  a. Plot a histogram of these data
  
  b. An engineering study previously done on heaters claims that these data should come from a Gamma distribution. If Gamma(a,b) stands for a Gamma distribution with mean a/b, what is the likelihood function of parameters a and b based on the data above?
  
  c. Plot the likelihood of a and b (plot the likelihood surface) over a grid of values for a and b
  
  d. Find the MLEs for a and b
  
  e. Given the ML estimates of a and b, find x