Chapter 4: Problem 26
Compute \(P\left(Y_{3}<\xi_{0.5}
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Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample drawn from a \(N\left(\mu, \sigma^{2}\right)\) distribution. As discussed in Example 4.2.1, the pivot random variable for a confidence interval is $$ t=\frac{\bar{X}-\mu}{S / \sqrt{n}} $$ where \(\bar{X}\) and \(S\) are the sample mean and standard deviation, respectively. Recall by Theorem \(3.6 .1\) that \(t\) has a Student \(t\) -distribution with \(n-1\) degrees of freedom; hence, its distribution is free of all parameters for this normal situation. In the notation of this section, \(t_{n-1}^{(\gamma)}\) denotes the \(\gamma 100 \%\) percentile of a \(t\) -distribution with \(n-1\) degrees of freedom. Using this notation, show that a \((1-\alpha) 100 \%\) confidence interval for \(\mu\) is $$ \left(\bar{x}-t^{(1-\alpha / 2)} \frac{s}{\sqrt{n}}, \bar{x}-t^{(\alpha / 2)} \frac{s}{\sqrt{n}}\right) $$
Let the result of a random experiment be classified as one of the mutually exclusive and exhaustive ways \(A_{1}, A_{2}, A_{3}\) and also as one of the mutually exhaustive ways \(B_{1}, B_{2}, B_{3}, B_{4}\). Say that 180 independent trials of the experiment result in the following frequencies: \begin{tabular}{|c|c|c|c|c|} \hline & \(B_{1}\) & \(B_{2}\) & \(B_{3}\) & \(B_{4}\) \\ \hline\(A_{1}\) & \(15-3 k\) & \(15-k\) & \(15+k\) & \(15+3 k\) \\ \hline\(A_{2}\) & 15 & 15 & 15 & 15 \\ \hline\(A_{3}\) & \(15+3 k\) & \(15+k\) & \(15-k\) & \(15-3 k\) \\ \hline \end{tabular} where \(k\) is one of the integers \(0,1,2,3,4,5\). What is the smallest value of \(k\) that leads to the rejection of the independence of the \(A\) attribute and the \(B\) attribute at the \(\alpha=0.05\) significance level?
Let \(Y_{1}
For \(\alpha>0\) and \(\beta>0\), consider the following accept-reject algorithm: 1\. Generate \(U_{1}\) and \(U_{2}\) iid uniform \((0,1)\) random variables. Set \(V_{1}=U_{1}^{1 / \alpha}\) and \(V_{2}=U_{2}^{1 / \beta}\) 2\. Set \(W=V_{1}+V_{2}\). If \(W \leq 1\), set \(X=V_{1} / W\); else go to step 1 . 3\. Deliver \(X\). Show that \(X\) has a beta distribution with parameters \(\alpha\) and \(\beta,(3.3 .9) .\) See Kennedy and Gentle (1980).
For the proof of Theorem 4.8.1, we assumed that the cdf was strictly
increasing over its support. Consider a random variable \(X\) with cdf \(F(x)\)
that is not strictly increasing. Define as the inverse of \(F(x)\) the function
$$
F^{-1}(u)=\inf \\{x: F(x) \geq u\\}, \quad 0
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