Of 70 restaurant bills, 40 of the 70 were paid using cash. We assume a random sample and estimate the posterior distribution of the binomial parameterp, the population proportion paying cash. LetXdenote the number of bills paid using cash out of 70 bills.

Answer the following quesitons.

1. Use a beta prior distribution witha= 3 andb= 3. Compute the posterior distribu-
2. tion ofp.
3. What is the posterior mean ofp,E(p|X= 40)?
4. ExpressE(p|X= 40) in terms of the prior mean ofpand the sample mean ofX.