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Chapter 2: Problem 9
Let \(X\) and \(Y\) have the joint \(\operatorname{pmf} p(x, y)=\frac{1}{7},(0,0),(1,0),(0,1),(1,1),(2,1)\), \((1,2),(2,2)\), zero elsewhere. Find the correlation coefficient \(\rho .\)
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Let \(f\left(x_{1}, x_{2}\right)=4 x_{1} x_{2}, 0
Let \(X_{1}\) and \(X_{2}\) be two independent random variables so that the variances of \(X_{1}\) and \(X_{2}\) are \(\sigma_{1}^{2}=k\) and \(\sigma_{2}^{2}=2\), respectively. Given that the variance of \(Y=3 X_{2}-X_{1}\) is 25, find \(k\)
Let \(X_{1}\) and \(X_{2}\) have the joint pmf \(p\left(x_{1}, x_{2}\right)\) described as follows: $$ \begin{array}{c|cccccc} \left(x_{1}, x_{2}\right) & (0,0) & (0,1) & (1,0) & (1,1) & (2,0) & (2,1) \\ \hline p\left(x_{1}, x_{2}\right) & \frac{1}{18} & \frac{3}{18} & \frac{4}{18} & \frac{3}{18} & \frac{6}{18} & \frac{1}{18} \end{array} $$ and \(p\left(x_{1}, x_{2}\right)\) is equal to zero elsewhere. Find the two marginal probability mass functions and the two conditional means. Hint: Write the probabilities in a rectangular array.
Suppose \(X_{1}\) and \(X_{2}\) have the joint pdf $$ f\left(x_{1}, x_{2}\right)=\left\\{\begin{array}{ll} e^{-x_{1}} e^{-x_{2}} & x_{1}>0, x_{2}>0 \\ 0 & \text { elsewhere } \end{array}\right. $$ For constants \(w_{1}>0\) and \(w_{2}>0\), let \(W=w_{1} X_{1}+w_{2} X_{2}\) (a) Show that the pdf of \(W\) is $$ f_{W}(w)=\left\\{\begin{array}{ll} \frac{1}{w_{1}-w_{2}}\left(e^{-w / w_{1}}-e^{-w / w_{2}}\right) & w>0 \\ 0 & \text { elsewhere } \end{array}\right. $$ (b) Verify that \(f_{W}(w)>0\) for \(w>0\). (c) Note that the pdf \(f_{W}(w)\) has an indeterminate form when \(w_{1}=w_{2}\). Rewrite \(f_{W}(w)\) using \(h\) defined as \(w_{1}-w_{2}=h\). Then use l'H么pital's rule to show that when \(w_{1}=w_{2}\), the pdf is given by \(f_{W}(w)=\left(w / w_{1}^{2}\right) \exp \left\\{-w / w_{1}\right\\}\) for \(w>0\) and zero elsewhere.
Let \(F(x, y)\) be the distribution function of \(X\) and \(Y\). For all real
constants \(a
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