Let X1, ..., Xn be a random sample of an Exponential population with parameter p. That is,
f(x|p) = pe-px , x > 0
Suppose we put a Gamma (c, d) prior on p.
Find the Bayes estimator of p if we use the loss function L(p, a) = (p - a)2.
Let X1, ..., Xn be a random sample of an Exponential population with parameter p, i.e.,
The likelihood function is
The prior distribution of p follows Gamma (c,d), i.e.,
Then, the posterior distribution is the product of likelihood and prior distribution
Under the squared loss function L(p, a) = (p - a)2 , Bayes estimator is
The integral is a gamma function and its density is equal to one when we multiple by some constant, consider
a = n+c+1
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