Statistics | Part IB, 2001

Suppose the single random variable XX has a uniform distribution on the interval [0,θ][0, \theta] and it is required to estimate θ\theta with the loss function

L(θ,a)=c(θa)2L(\theta, a)=c(\theta-a)^{2}

where c>0c>0.

Find the posterior distribution for θ\theta and the optimal Bayes point estimate with respect to the prior distribution with density p(θ)=θeθ,θ>0p(\theta)=\theta e^{-\theta}, \theta>0.

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