Chapter 3: Problem 11
Show that the \(t\) -distribution with \(r=1\) degree of freedom and the Cauchy distribution are the same.
Chapter 3: Problem 11
Show that the \(t\) -distribution with \(r=1\) degree of freedom and the Cauchy distribution are the same.
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Get started for freeLet \(X_{1}, X_{2}, X_{3}\) be iid random variables each having a standard normal distribution. Let the random variables \(Y_{1}, Y_{2}, Y_{3}\) be defined by $$ X_{1}=Y_{1} \cos Y_{2} \sin Y_{3}, \quad X_{2}=Y_{1} \sin Y_{2} \sin Y_{3}, \quad X_{3}=Y_{1} \cos Y_{3} $$ where \(0 \leq Y_{1}<\infty, 0 \leq Y_{2}<2 \pi, 0 \leq Y_{3} \leq \pi .\) Show that \(Y_{1}, Y_{2}, Y_{3}\) are mutually independent.
A certain job is completed in three steps in series. The means and standard deviations for the steps are (in minutes): $$ \begin{array}{ccc} \hline \text { Step } & \text { Mean } & \text { Standard Deviation } \\ \hline 1 & 17 & 2 \\ 2 & 13 & 1 \\ 3 & 13 & 2 \\ \hline \end{array} $$Assuming independent steps and normal distributions, compute the probability that the job will take less than 40 minutes to complete.
. Let \(X\) and \(Y\) have the joint \(\operatorname{pmf} p(x, y)=e^{-2} /[x !(y-x) !], y=0,1,2, \ldots ;\) \(x=0,1, \ldots, y\), zero elsewhere. (a) Find the mgf \(M\left(t_{1}, t_{2}\right)\) of this joint distribution. (b) Compute the means, the variances, and the correlation coefficient of \(X\) and \(Y\). (c) Determine the conditional mean \(E(X \mid y)\). Hint: Note that $$ \sum_{x=0}^{y}\left[\exp \left(t_{1} x\right)\right] y ! /[x !(y-x) !]=\left[1+\exp \left(t_{1}\right)\right]^{y} $$ Why?
Let \(X\) and \(Y\) have a bivariate normal distribution with parameters \(\mu_{1}=\) 20, \(\mu_{2}=40, \sigma_{1}^{2}=9, \sigma_{2}^{2}=4\), and \(\rho=0.6\). Find the shortest interval for which \(0.90\) is the conditional probability that \(Y\) is in the interval, given that \(X=22\).
Let \(F\) have an \(F\) -distribution with parameters \(r_{1}\) and \(r_{2} .\) Prove that \(1 / F\) has an \(F\) -distribution with parameters \(r_{2}\) and \(r_{1}\).
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