STAT 4198 Homework 4 Spring 2023
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
STAT 4198
Homework 4
Spring 2023
1. The numbers of misprints spotted on the first few pages of an early draft of a book are: 3, 4, 2, 1, 2, 3.
It seems reasonable that these numbers would constitute a sample from a Poisson distribution of unknown mean 入.
a) Suppose you have no knowledge of the typist’s skills. Use the following uniform prior: define 100 hypotheses where 入 is 0. 1, 0.2, …, 10, and is equally likely to be any ofthese values.
i) Calculate the posterior distribution of 入. Plot this posterior.
ii) What is the mean of this posterior distribution?
iii) Give a 95% credible interval for 入.
iv) What is the probability that 入 is greater than 3?
b) Suppose you have no knowledge of the typist’s skills, but instead of a discrete uniform prior you use an uniformative continuous Gamma(0.01, 0.01) prior for 入. Repeat part
a) with this prior.
c) Now suppose the typist thinks that we should use a prior with a mean of 3, and a standard deviation of 2. Use a suitable Gamma prior and repeat part a).
d) Plot the probability mass function for the number of errors on a randomly selected page. Do this via Monte Carlo simulation, using your posterior from part c).
2. A manufacturer is interested in the time to failure of his batteries.
Suppose the time to failure of the batteries has an exponential distribution:
p(x|e) = ee−ex
Note that the mean of this population is 1/e.
The manufacturer is interested in estimating e. Suppose that n batteries are randomly selected and let their failure times be x1, x2,…, xn .
Suppose we use a gamma prior for e.
p(e|a, F) = 
e a−1e −Fe
a) What is the posterior distribution of e? Is the Gamma a conjugate prior for an exponential likelihood?
b) Suppose an expert believes that the mean time to failure is 120 hours. Furthermore, he claims that he is 95% sure that the mean time to failure is between 100 and 150 hours.
What would be a suitable choice for a and F in the prior?
[Hint: For a Gamma(a , F) random variable
Mean = 
and Variance = 
]
c) Suppose a sample of 5 batteries are taken and the failure times are:
113, 130, 128, 128, 141
i) What is the posterior distribution of e? (using your prior from part b))
ii) What is the mean and variance of this posterior distribution? Give a 95% credible interval for e. Give a 95% credible interval for 1/e.
2023-04-07