6. Suppose the common pmf of the random sample X₁, X₂, ..., X, is Poisson with unknown parameter > 0, with pmf: -0 0 e " x! x = 0, 1, 2, Suppose also that the prior distribution of the parameter 0 is gamma with pdf: a-1 -- e a > 0, λ > 0. f(x|0) = 1 г(α)λª ƒ(0|a, λ) = - " A (a) Find a Bayesian Estimator (BE) 0 of 0. B ^ (b) Express from part (a) as a function of the sample mean X and the mean λ of 0. B

If the number X of particles emitted during a 1-hour period from a radioactive source has a poisson distribution with parameter equal to 4 and that the probability that any emitted is recorded is p=0.9 find the probability distribution of the number Y of the particles recorded in a 1-hour and hence the probability that no particle is recorded

Suppose that Z1, Z2, . . . , Zn are statistically independent random variables. Define Y as the sum of squares of these random variables

1.) Consider the discrete random variable with pmf: P(X=x)=(A/(4^x)), x=2,3,4,5,.... What is the value of A?