Chapter 5: Problem 16
Let \(Y_{1}
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chapter 5: Problem 16
Let \(Y_{1}
These are the key concepts you need to understand to accurately answer the question.
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Get started for freeLet \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a \(\Gamma(1, \beta)\) distribution. (a) Show that the confidence interval \(\left(2 n \bar{X} /\left(\chi_{2 n}^{2}\right)^{(1-(\alpha / 2))}, 2 n \bar{X} /\left(\chi_{2 n}^{2}\right)^{(\alpha / 2)}\right)\) is an exact \((1-\alpha) 100 \%\) confidence interval for \(\beta\). (b) Show that value of a \(90 \%\) confidence interval for the data of the example is \((64.99,136.69)\)
. 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, $$ \begin{array}{lllll} 44.8079 & 1.5215 & 12.1929 & 12.5734 & 43.2305 \end{array} $$
Let \(Y_{2}\) and \(Y_{n-1}\) denote the second and the \((n-1)\) st order statistics of a random sample of size \(n\) from a distribution of the continuous type having a distribution function \(F(x)\). Compute \(P\left[F\left(Y_{n-1}\right)-F\left(Y_{2}\right) \geq p\right]\), where \(0
Let \(Y_{1}
. Let \(Y_{1}
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