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let X, Y be random variables. Also let X|Y = y ~ Poisson(y) and Y ~...

let X, Y be random variables. Also let X|Y = y ~ Poisson(y) and Y ~ gamma(a,b) is the prior distribution for Y. a and b are also known.

1. Find the posterior distribution of Y|X=x where X=(X1, X2, ... , Xn) and x is an observed sample of size n from the distribution of X.

2. Suppose the number of people who visit a nursing home on a day is Poisson random variable and the parameter of the Poisson distribution has a gamma(4,20) distribution. On a Saturday, 83 people visited the nursing home. Use part 1. to derive the posterior mean of Y|X=x and then compute its value using the info in part 2.  

*the sum of independent Poisson random variables has a Poisson distribution

Math   Statistics-And-Probability   
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lastly let me tell you that if my answer does not match with that of yours its because of the structure of gamma distribution that I have taken.All you need to do is just interchange the parameters of gamma distribution and proceed.The procedure is perfect I assure you that.

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