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 \(f(x, y)=e^{-x-y}, 0
Let the random variables \(X\) and \(Y\) have the joint pmf (a) \(p(x, y)=\frac{1}{3},(x, y)=(0,0),(1,1),(2,2)\), zero elsewhere. (b) \(p(x, y)=\frac{1}{3},(x, y)=(0,2),(1,1),(2,0)\), zero elsewhere. (c) \(p(x, y)=\frac{1}{3},(x, y)=(0,0),(1,1),(2,0)\), zero elsewhere. In each case compute the correlation coefficient of \(X\) and \(Y\).
If the correlation coefficient \(\rho\) of \(X\) and \(Y\) exists, show that \(-1 \leq \rho \leq 1\). Hint: Consider the discriminant of the nonnegative quadratic function $$h(v)=E\left\\{\left[\left(X-\mu_{1}\right)+v\left(Y-\mu_{2}\right)\right]^{2}\right\\}$$ where \(v\) is real and is not a function of \(X\) nor of \(Y\).
Let \(\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}\) 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 \(X_{1}, X_{2}, X_{3}\) be iid with common pdf \(f(x)=\exp (-x), 0
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