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 the random variables \(X_{1}\) and \(X_{2}\) have the joint pdf \(f\left(x_{1}, x_{2}\right)=1 / \pi\), for \(\left(x_{1}-1\right)^{2}+\left(x_{2}+2\right)^{2}<1\), zero elsewhere. Find \(f_{1}\left(x_{1}\right)\) and \(f_{2}\left(x_{2}\right) .\) Are \(X_{1}\) and \(X_{2}\) independent?
If \(p\left(x_{1}, x_{2}\right)=\left(\frac{2}{3}\right)^{x_{1}+x_{2}}\left(\frac{1}{3}\right)^{2-x_{1}-x_{2}},\left(x_{1}, x_{2}\right)=(0,0),(0,1),(1,0),(1,1)\), zero elsewhere, is the joint pmf of \(X_{1}\) and \(X_{2}\), find the joint pmf of \(Y_{1}=X_{1}-X_{2}\) and \(Y_{2}=X_{1}+X_{2}\)
Let \(F(x, y)\) be the distribution function of \(X\) and \(Y .\) For all real constants \(a
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 density functions and the two conditional means.
Let \(X_{1}, X_{2}, X_{3}\) be iid with common pdf \(f(x)=e^{-x}, x>0,0\) elsewhere. Find the joint pdf of \(Y_{1}=X_{1} / X_{2}, Y_{2}=X_{3} /\left(X_{1}+X_{2}\right)\), and \(Y_{3}=X_{1}+X_{2}\). Are \(Y_{1}, Y_{2}, Y_{3}\) mutually independent?
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