2.7 ( ) Consider a binomial random variable x given by (2.9), with prior distribution for given by the beta distribution (2.13), and suppose

2.7 ( ) Consider a binomial random variable x given by (2.9), with prior distribution for μ given by the beta distribution (2.13), and suppose we have observed m occurrences of x = 1and l occurrences of x = 0. Show that the posterior mean value of x lies between the prior mean and the maximum likelihood estimate for μ. To do this, show that the posterior mean can be written as λ times the prior mean plus (1 − λ)

times the maximum likelihood estimate, where 0
λ
1. This illustrates the concept of the posterior distribution being a compromise between the prior distribution and the maximum likelihood solution.