Chapter 3: Problem 9
Toss two nickels and three dimes at random. Make appropriate assumptions and compute the probability that there are more heads showing on the nickels than on the dimes.
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Using the computer, obtain plots of beta pdfs for \(\alpha=5\) and \(\beta=1,2,5,10,20\).
. Let \(X\) and \(Y\) have a bivariate normal distribution with respective
parameters \(\mu_{x}=2.8, \mu_{y}=110, \sigma_{x}^{2}=0.16,
\sigma_{y}^{2}=100\), and \(\rho=0.6 .\) Compute:
(a) \(P(106
For the Burr distribution, show that $$ E\left(X^{k}\right)=\frac{1}{\beta^{k / \tau}} \Gamma\left(\alpha-\frac{k}{\tau}\right) \Gamma\left(\frac{k}{\tau}+1\right) / \Gamma(\alpha) $$ provided \(k<\alpha \tau\)
Let \(X_{1}, X_{2}, X_{3}\) 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.
. Using the computer, obtain plots of the pdfs of chi-squared distributions with degrees of freedom \(r=1,2,5,10,20\). Comment on the plots.
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