Question: Let an observed random variable Z depend on a parameter λ according to the conditional pdf The α priori pdf of λ is where β

Let an observed random variable Z depend on a parameter λ according to the conditional pdf z > 0, 1 > 0 z < 0 fZIA (z | 1) = 0,

The α priori pdf of λ is

рт -е-ра ят-1, 120 fA (4) = Г(т) 0,

where β and m  are parameters and Γ(m) is the gamma function. Assume that m is a positive integer. 

(a) Find E{λ} and var {λ} before any observations are made; that is, find the mean and variance of λ using fΛ (λ).

(b) Assume one observation is made. Find fΛ|Z (λ|z1) and hence the minimum mean squared error (conditional-mean) estimate of λ and the variance of the estimate. Compare with part (a). Comment on the similarity of fΛ (λ) and fΛ|Z (λ|z1).

(c) Making use of part (b), find the posterior pdf of λ given two observations fΛ|Z (λ|z1, z2). Find the minimum mean-squared error estimate of λ based on two observations and its variance. Compare with parts (a) and (b), and comment.

(d) Generalize the preceding to the case in which K observations are used to estimate λ.

z > 0, 1 > 0 z < 0 fZIA (z | 1) = 0, -- -1, 120 fA (4) = () 0,

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