Suppose that X1,..X, is a random sample from a Poisson distribution with mean.. a) Show that the conjugate prior family of distributions for is the gamma family of distributions. b) A scientist...

Suppose that X1,..X, is a random sample from a Poisson distribution with mean..
a) Show that the conjugate prior family of distributions for is the gamma family of distributions.
b) A scientist believes that the number of times that a certain event occurs in a small
interval of time follows a Poisson distribution. Furthermore she believes that the mean
of this distribution is approximately 5 and she is 99% sure that the mean is less than 10. Find the conjugate prior distribution which best captures the scientists prior beliefs. You may restrict your search for the hyper parameter a to integers c) Based on the observed data below and the prior you found in part (c), determine the posterior mean and variance of the posterior probability that < 10 and construct an equitailed 95% posterior credible interval for 10="" and="" construct="" an="" equitailed="" 95%="" posterior="" credible="" interval="">