Chapter 7: Problem 2
Prove that the sum of the observations of a random sample of size \(n\) from a Poisson distribution having parameter \(\theta, 0<\theta<\infty\), is a sufficient statistic for \(\theta\).
Chapter 7: Problem 2
Prove that the sum of the observations of a random sample of size \(n\) from a Poisson distribution having parameter \(\theta, 0<\theta<\infty\), is a sufficient statistic for \(\theta\).
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Get started for freeLet \(X_{1}, X_{2}, \ldots, X_{n}\) 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 \(Y_{1}
We consider a random sample \(X_{1}, X_{2}, \ldots, X_{n}\) from a distribution
with pdf \(f(x ; \theta)=(1 / \theta) \exp (-x / \theta), 0
Write the pdf
$$f(x ; \theta)=\frac{1}{6 \theta^{4}} x^{3} e^{-x / \theta}, \quad
0
Let \(\bar{X}\) denote the mean of the random sample \(X_{1}, X_{2}, \ldots, X_{n}\) from a gammatype distribution with parameters \(\alpha>0\) and \(\beta=\theta \geq 0 .\) Compute \(E\left[X_{1} \mid \bar{x}\right]\). Hint: Can you find directly a function \(\psi(\bar{X})\) of \(\bar{X}\) such that \(E[\psi(X)]=\theta ?\) Is \(E\left(X_{1} \mid \bar{x}\right)=\psi(\bar{x}) ?\) Why?
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