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Chapter 9: Problem 481
Given the values \(4,4,6,7,9\) give the deviation of each from the mean.
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Let \(\mathrm{X}\) have the probability distribution defined by $$ \begin{array}{lcc} \mathrm{f}(\mathrm{x})=1-\mathrm{e}^{-\mathrm{x}} & \text { for } & \mathrm{x} \geq 0 \\ \text { and }=0 & \text { for } & \mathrm{x}<0 . \end{array} $$ Let \(\mathrm{Y}=\sqrt{\mathrm{X}}\) be a new random variable. Find \(\mathrm{G}(\mathrm{y})\), the distribution function of \(\mathrm{Y}\), using the cumulative distribution function technique.
Given a normal population with \(\mu=25\) and \(\sigma=5\), find the probability that an assumed value of the variable will fall in the interval 20 to 30 .
Let \(\mathrm{X}_{1}, \ldots, \mathrm{X}_{\mathrm{n}}\) be a random sample from a normal distribution with mean \(\mu\) and variance \(\sigma^{2} .\) Let \(\left(0_{1}, 0_{2}\right)=(\mu / \sigma) .\) Estimate the parameters \(\mu\) and \(\sigma\) by the method of moments.
\(\mathrm{X}\) is a discrete random variable with probability mass function, \(\mathrm{f}(\mathrm{x})=1 / \mathrm{n}, \mathrm{x}=1,2,3, \ldots, \mathrm{n} ;=0\) otherwise. If \(\mathrm{Y}=\mathrm{X}^{2}\), find the probability mass function of \(\mathrm{Y}\).
Consider the following situation: A normal distribution of a random variable, \(\mathrm{X}\), has a variance \(\sigma_{1}^{2}\), where \(\sigma_{1}^{2}\) is unknown. It is found however that experimental values of \(\mathrm{X}\) have a wide dispersion indicating that \(\sigma_{1}^{2}\) must be quite large. A certain modification in the experiment is made to reduce the variance. Let the post-modification random variable be denoted \(\mathrm{Y}\), and let \(\mathrm{Y}\) have a normal distribution with variance \(\sigma_{2}^{2}\). Find a completely general method of determining confidence intervals for ratios of variances, \(\sigma_{1}^{2} / \sigma_{2}^{2}\)
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