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 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 f_Λ (λ).

(b) Assume one observation is made. Find f_Λ|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 f_Λ (λ) and f_Λ|Z (λ|z₁).

(c) Making use of part (b), find the posterior pdf of λ given two observations f_Λ|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.

Transcribed Image Text:

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