Chapter 2: Problem 8
Let \(X\) and \(Y\) have the joint pdf \(f(x, y)=3 x, 0
Chapter 2: Problem 8
Let \(X\) and \(Y\) have the joint pdf \(f(x, y)=3 x, 0
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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?
A fair die is cast at random three independent times. Let the random variable \(X_{i}\) be equal to the number of spots that appear on the \(i\) th trial, \(i=1,2,3\). Let the random variable \(Y\) be equal to \(\max \left(X_{i}\right) .\) Find the cdf and the pmf of \(Y\). Hint: \(P(Y \leq y)=P\left(X_{i} \leq y, i=1,2,3\right)\).
Two line segments, each of length two units, are placed along the \(x\) -axis. The midpoint of the first is between \(x=0\) and \(x=14\) and that of the second is between \(x=6\) and \(x=20 .\) Assuming independence and uniform distributions for these midpoints, find the probability that the line segments overlap.
Let \(X_{1}, X_{2}, X_{3}\) be iid with common mgf \(M(t)=\left((3 / 4)+(1 / 4) e^{t}\right)^{2}\), for all \(t \in R\) (a) Determine the probabilities, \(P\left(X_{1}=k\right), k=0,1,2\). (b) Find the mgf of \(Y=X_{1}+X_{2}+X_{3}\) and then determine the probabilities, \(P(Y=k), k=0,1,2, \ldots, 6\)
Find the probability of the union of the events
\(a
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