Let y 1 , y 2 , y 3 , , y n , represent a random sample of size n selected from a Poisson probability distribution with unknown mean Let X represent the sum of the Poisson values, X Y i Then X has a Poisson distribution with mean n Assume that the prior distribution for is an exponential probability distribution with parameter a Find the posterior distribution, g( x) b Using a squared error loss function, find the Bayesian estimate of

Question: Let y 1 , y 2 , y 3 , . . . , y n , represent a random sample of size n selected

Let y₁, y₂, y₃, . . . , y_n, represent a random sample of size n selected from a Poisson probability distribution with unknown mean λ. Let X represent the sum of the Poisson values, X = ∑Y_i. Then X has a Poisson distribution with mean nλ. Assume that the prior distribution for λ is an exponential probability distribution with parameter β.
a. Find the posterior distribution, g(λ Ι x).
b. Using a squared error loss function, find the Bayesian estimate of λ.

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