In this problem, we revisit the example that we worked on in class about Kevin Mitchell's batting average,

and we imagine that it is $1988$ now! Assuming that the outcomes of Kevin Mitchell's hits are independent,

the number of his home runs, call it $x,$ has the following distribution:

$xBinomial(N,),$

where $N$ is the number of times he is at bat, and $$ is his batting average $($the probability that he makes a

home$-$run each time he is at bat$).$ We assume that $$ has a beta prior distribution with parameters $=2$ and

$=5.$ Using the Bayesian framework, our goals is to obtain a distribution for $,$ his probability of making a

home run. Assume that we have observed him in his first $10$ games of the season where he was at bat $44$

times, and he had made $4$ home runs. What will be different than what we did in class is that we will use a

different prior for $.$ Specifically, we use the following shifted and truncated double exponential prior:

$f()=\frac{50}{1-\frac{e^{-95}+e^{-5}}{2}}$exp$\{-100|-0\mathrm{.}05|\},01.$

Problem $4($a$)$ points