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},
0
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Get started for freeLet \(X_{1}, X_{2}, X_{3}\) be iid, each with the distribution having pdf \(f(x)=e^{-x}, 0<\) \(x<\infty\), zero elsewhere. Show that $$Y_{1}=\frac{X_{1}}{X_{1}+X_{2}}, \quad Y_{2}=\frac{X_{1}+X_{2}}{X_{1}+X_{2}+X_{3}}, \quad Y_{3}=X_{1}+X_{2}+X_{3}$$ are mutually independent.
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 \(Y_{1}=X_{1}, Y_{2}=X_{1}+X_{2}\), and \(Y_{3}=X_{1}+X_{2}+X_{3}\)
Let \(f\left(x_{1}, x_{2}\right)=4 x_{1} x_{2}, 0
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 \(X\) and \(Y\) have the joint pdf \(f(x, y)=6(1-x-y), x+y<1,0
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