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let-x-1-dots-x-n-be-a-random-sample-from-bernoulli-theta-where-0-lt-theta-lt-1-us-unknow-pa605

Let

`x_(1),dots,x_(n)`

be a random sample from Bernoulli

`(\theta )`

, where

`0<\theta <1`

us unknown. Suppose it is known that

`\theta ?`

Uniform

`(0,1)`

. Find the Bayesian estimator for

`\theta `

. Solution: The prior distribution is We also know that

`T=`

is sufficient for

`\theta `

(show by FNFT) and that

`T?`

(show by mgf technique) Thus, the posterior distribution is