Chapter 2: Problem 2
Let \(f_{1 \mid 2}\left(x_{1} \mid x_{2}\right)=c_{1} x_{1} / x_{2}^{2},
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Chapter 2: Problem 2
Let \(f_{1 \mid 2}\left(x_{1} \mid x_{2}\right)=c_{1} x_{1} / x_{2}^{2},
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Get started for freeSuppose \(X_{1}\) and \(X_{2}\) are discrete random variables which have the joint pmf \(p\left(x_{1}, x_{2}\right)=\left(3 x_{1}+x_{2}\right) / 24,\left(x_{1}, x_{2}\right)=(1,1),(1,2),(2,1),(2,2)\), zero elsewhere. Find the conditional mean \(E\left(X_{2} \mid x_{1}\right)\), when \(x_{1}=1\)
Let \(X\) and \(Y\) be random variables with the space consisting of the four points: \((0,0),(1,1),(1,0),(1,-1)\). Assign positive probabilities to these four points so that the correlation coefficient is equal to zero. Are \(X\) and \(Y\) independent?
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 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 \(Y\).
Find \(P\left(0
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