Chapter 2: Problem 6
Let \(X\) and \(Y\) have the joint pdf \(f(x, y)=1,-x
Chapter 2: Problem 6
Let \(X\) and \(Y\) have the joint pdf \(f(x, y)=1,-x
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Get started for freeLet \(\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}\) be the common variance of \(X_{1}\) and \(X_{2}\) and let \(\rho\) be the correlation coefficient of \(X_{1}\) and \(X_{2}\). Show that $$P\left[\left|\left(X_{1}-\mu_{1}\right)+\left(X_{2}-\mu_{2}\right)\right| \geq k \sigma\right] \leq \frac{2(1+\rho)}{k^{2}}$$
Let \(X\) and \(Y\) have the joint pmf described as follows: $$\begin{array}{c|cccccc}(x, y) & (1,1) & (1,2) & (1,3) & (2,1) & (2,2) & (2,3) \\ \hline p(x, y) & \frac{2}{15} & \frac{4}{15} & \frac{3}{15} & \frac{1}{15} & \frac{1}{15} & \frac{4}{15} \end{array}$$ and \(p(x, y)\) is equal to zero elsewhere. (a) Find the means \(\mu_{1}\) and \(\mu_{2}\), the variances \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\), and the correlation coefficient \(\rho\). (b) Compute \(E(Y \mid X=1), E\left(Y \mid X=2\right.\) ), and the line \(\mu_{2}+\rho\left(\sigma_{2} / \sigma_{1}\right)\left(x-\mu_{1}\right) .\) Do the points \([k, E(Y \mid X=k)], k=1,2\), lie on this line?
Let \(A_{1}=\\{(x, y): x \leq 2, y \leq 4\\}, A_{2}=\\{(x, y): x \leq 2, y \leq
1\\}, A_{3}=\)
\(\\{(x, y): x \leq 0, y \leq 4\\}\), and \(A_{4}=\\{(x, y): x \leq 0 y \leq
1\\}\) be subsets of the
space \(\mathcal{A}\) of two random variables \(X\) and \(Y\), which is the entire
two-dimensional plane. If \(P\left(A_{1}\right)=\frac{7}{8},
P\left(A_{2}\right)=\frac{4}{8}, P\left(A_{3}\right)=\frac{3}{8}\), and
\(P\left(A_{4}\right)=\frac{2}{8}\), find \(P\left(A_{5}\right)\), where
\(A_{5}=\\{(x, y): 0
Let \(X_{1}, X_{2}, X_{3}, X_{4}\) have the joint pdf \(f\left(x_{1}, x_{2},
x_{3}, x_{4}\right)=24,0
Use the formula \((2.2 .1)\) to find the pdf of \(Y_{1}=X_{1}+X_{2}\), where
\(X_{1}\) and \(X_{2}\) have the joint pdf \(f x_{1}, x_{2}\left(x_{1},
x_{2}\right)=2 e^{-\left(x_{1}+x_{2}\right)}, 0
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