91Ó°ÊÓ

Write the pdf $$ f(x ; \theta)=\frac{1}{6 \theta^{4}} x^{3} e^{-x / \theta}, \quad 0

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a distribution that is \(N(\theta, 1),-\infty<\theta<\infty\). Find the MVUE of \(\theta^{2}\). Hint: \(\quad\) First determine \(E\left(\bar{X}^{2}\right)\).

Show that the mean \(\bar{X}\) of a random sample of size \(n\) from a distribution having pdf \(f(x ; \theta)=(1 / \theta) e^{-(x / \theta)}, 0

Let \(Y_{1}

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from each of the following distributions involving the parameter \(\theta .\) In each case find the mle of \(\theta\) and show that it is a sufficient statistic for \(\theta\) and hence a minimal sufficient statistic. (a) \(b(1, \theta)\), where \(0 \leq \theta \leq 1\). (b) Poisson with mean \(\theta>0\). (c) Gamma with \(\alpha=3\) and \(\beta=\theta>0\). (d) \(N(\theta, 1)\), where \(-\infty<\theta<\infty\). (e) \(N(0, \theta)\), where \(0<\theta<\infty\)

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be iid \(N(0, \theta), 0<\theta<\infty\). Show that \(\sum_{1}^{n} X_{i}^{2}\) is_a sufficient statistic for \(\theta\).

Let \(Y_{1}

Let \(Y_{1}

Let \(Y_{1}

Show that each of the following families is not complete by finding at least one nonzero function \(u(x)\) such that \(E[u(X)]=0\), for all \(\theta>0\). (a) $$ f(x ; \theta)=\left\\{\begin{array}{ll} \frac{1}{2 \theta} & -\theta