Bayesian Inference - website shutting down

There is a famous website but it sometimes suffers from DDoS attacks so it goes down sometimes, We assume that the numbers of shutting down of the website in the duration of one month has Poisson() distribution

c) It happens before so you should have some experience, hence you believe that in average the number of website is down is 4. So propose a good Gamma(, ) prior () with the mean of 4 and SD = 2 (standard deviation)

d) We know that the gamma distribution is a conjugate for the Poisson distribution, and by using the i.i.d from last five months we know that x' = (4, 1, 5, 3, 2) then calculate a posterior, and find the maximum a posteriori estimate for ?

It's clear that X - Gamma(, ) random variable has

a mean / ,

standard deviation /

and mode (1)/ (for 1)