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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\(a
Show that the function \(F(x, y)\) that is equal to 1 provided that \(x+2 y \geq 1\), and that is equal to zero provided that \(x+2 y<1\), cannot be a distribution function of two random variables.
Let \(X_{1}, X_{2}\) be two random variables with joint \(\mathrm{pmf} p\left(x_{1}, x_{2}\right)=\left(x_{1}+x_{2}\right) / 12\), for \(x_{1}=1,2, x_{2}=1,2\), zero elsewhere. Compute \(E\left(X_{1}\right), E\left(X_{1}^{2}\right), E\left(X_{2}\right), E\left(X_{2}^{2}\right)\), and \(E\left(X_{1} X_{2}\right) .\) Is \(E\left(X_{1} X_{2}\right)=E\left(X_{1}\right) E\left(X_{2}\right) ?\) Find \(E\left(2 X_{1}-6 X_{2}^{2}+7 X_{1} X_{2}\right)\)
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 \(X_{1}, X_{2}, X_{3}\), and \(X_{4}\) be four independent random variables,
each with pdf \(f(x)=3(1-x)^{2}, 0
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