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(Solved): x in cm n taken from RSA is given by f(x|\theta ,n)={((2n)/(\theta ^(2n))x^(2n-1) if 0<=x<=\th ...
x in cm n taken from RSA is given by
f(x|\theta ,n)={((2n)/(\theta ^(2n))x^(2n-1) if 0<=x<=\theta ),(0 otherwise ):}
Furthermore, suppose that the prior distribution of \theta is uniform with probability density function
\pi (\theta )={((1)/(10) if 0<\theta <=10),(0 otherwise ):}
(a) Prove or disprove that the posterior probability density function of \theta is
h(\theta |x)={((10^(2n)(2n-1)x^(2n))/((10^(2n)x-10x^(2n))\theta ^(2n)) if x<=\theta <=10),(0 otherwise ):}
(b) If x=1, what is the mean, the mode and the variance of the posterior distribution of \theta ?
Do not simplify the variance of the posterior distribution.
(c) Suppose that n=1, and let hat(\theta ) be the mode of the posterior distribution of \theta given x.
Calculate the squared loss risk function for widehat(\theta ).