Chapter 5: Problem 4
. Let \(X_{1}, \ldots, X_{n}\) be a random sample from the \(\Gamma(2, \theta)\) distribution, where \(\theta\) is unknown. Let \(Y=\sum_{i=1}^{n} X_{i}\) (a) Find the distribution of \(Y\) and determine \(c\) so that \(c Y\) is an unbiased estimator of \(\theta\). (b) If \(n=5\), show that $$ P\left(9.59<\frac{2 Y}{\theta}<34.2\right)=0.95 $$ (c) Using Part (b), show that if \(y\) is the value of \(Y\) once the sample is drawn, then the interval $$ \left(\frac{2 y}{34.2}, \frac{2 y}{9.59}\right) $$ is a \(95 \%\) confidence interval for \(\theta\). (d) Suppose the sample results in the values, $$ \begin{array}{lllll} 44.8079 & 1.5215 & 12.1929 & 12.5734 & 43.2305 \end{array} $$
Short Answer
Step by step solution
Key Concepts
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