Chapter 5: Problem 8
Let \(Y_{1}
Chapter 5: Problem 8
Let \(Y_{1}
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Get started for freeLet \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{m}\) be two independent random samples from the respective normal distributions \(N\left(\mu_{1}, \sigma_{1}^{2}\right)\) and \(N\left(\mu_{2}, \sigma_{2}^{2}\right)\), where the four parameters are unknown. To construct a confidence interval for the ratio, \(\sigma_{1}^{2} / \sigma_{2}^{2}\), of the variances, form the quotient of the two independent chi-square variables, each divided by its degrees of freedom, namely $$ F=\frac{\frac{(m-1) S_{2}^{2}}{\sigma_{2}^{2}} /(m-1)}{\frac{(n-1) S_{1}^{2}}{\sigma_{1}^{2}} /(n-1)}=\frac{S_{2}^{2} / \sigma_{2}^{2}}{S_{1}^{2} / \sigma_{1}^{2}} $$
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a continuous type
distribution.
(a) Find \(P\left(X_{1} \leq X_{2}\right), P\left(X_{1} \leq X_{2}, X_{1} \leq
X_{3}\right), \ldots, P\left(X_{1} \leq X_{i}, i=2,3, \ldots, n\right)\)
(b) Suppose the sampling continues until \(X_{1}\) is no longer the smallest
observation, (i.e., \(\left.X_{j}
Consider the following algorithm: (1) Generate \(U\) and \(V\) independent uniform \((-1,1)\) random variables. (2) Set \(W=U^{2}+V^{2}\). (3) If \(W>1\) goto Step (1). (4) Set \(Z=\sqrt{(-2 \log W) / W}\) and let \(X_{1}=U Z\) and \(X_{2}=V Z\).
Let \(Y_{1}
A Weibull distribution with pdf
$$
f(x)=\left\\{\begin{array}{ll}
\frac{1}{\theta^{3}} 3 x^{2} e^{-x^{3} / \theta^{3}} & 0
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