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Chapter 2: Problem 7

Let \(X\) and \(Y\) have the pdf \(f(x, y)=1,0

Short Answer

Expert verified
The CDF \(F_Z(z)\) for \(Z = XY\) is \(z - \frac{z^2}{2}\) for \(0 < z < 1\), and is 1 when \(z \geq 1\). The PDF \(f_Z(z)\) of \(Z = XY\) is \(1 - z\) for \(0 < z < 1\), and is 0 when \(z \geq 1\).

Step by step solution

01

Identify individual random variables' properties

From the problem, it's given that \(X\) and \(Y\) are uniform random variables over the interval (0,1). The PDF is given as \(f(x, y)=1\) for \(0<x<1,0<y<1\), and zero elsewhere.
02

Define and find CDF of Z

Let \(Z = XY\). We need to find the CDF and PDF of \(Z\). The CDF is defined as \(F_Z(z) = P(Z 鈮� z) = P(XY 鈮� z)\). In this case, we solve for \(P(XY 鈮� z)\) for the limit of \(Z\) which would lie in the interval (0,1) as \(X \in (0,1)\) and \(Y \in (0,1)\) and \(Z=XY\). Thus, \(P(XY 鈮� z) = \int_{0}^{1}\int_{0}^{z/y} 1 dx dy\), as long as \(0 < z < 1\), and is equal to 1, when \(z \geq 1\). Solving the double integral gives \(F_Z(z) = z - \frac{z^2}{2}\) for \(0 < z < 1\), and 1 when \(z \geq 1\).
03

Find PDF of Z

The PDF \(f_Z(z)\) of \(Z\) is the derivative of the CDF \(F_Z(z)\). That is \(f_Z(z) = dF_Z(z) / dz\). Differentiate the CDF determined in the previous step to obtain the PDF. Doing so, we get \(f_Z(z) = 1 - z\) for \(0 < z < 1\), and 0 when \(z \geq 1\).

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Most popular questions from this chapter

Let \(X_{1}, X_{2}\) have the joint pdf \(f\left(x_{1}, x_{2}\right)=1 / \pi, 0

In each case compute the correlation coefficient of \(X\) and \(Y\). Let \(X\) and \(Y\) have the joint pmf described as follows: \begin{tabular}{c|cccccc} \((x, y)\) & \((1,1)\) & \((1,2)\) & \((1,3)\) & \((2,1)\) & \((2,2)\) & \((2,3)\) \\ \hline\(p(x, y)\) & \(\frac{2}{15}\) & \(\frac{4}{15}\) & \(\frac{3}{15}\) & \(\frac{1}{15}\) & \(\frac{1}{15}\) & \(\frac{4}{15}\) \end{tabular} and \(p(x, y)\) is equal to zero elsewhere. (a) Find the means \(\mu_{1}\) and \(\mu_{2}\), the variances \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\), and the correlation coefficient \(\rho\). (b) Compute \(E(Y \mid X=1), E(Y \mid X=2)\), and the line \(\mu_{2}+\rho\left(\sigma_{2} / \sigma_{1}\right)\left(x-\mu_{1}\right) .\) Do the points \([k, E(Y \mid X=k)], k=1,2\), lie on this line?

Let \(X_{1}, X_{2}, X_{3}\) be iid with common pdf \(f(x)=\exp (-x), 0

Let \(X, Y, Z\) have joint pdf \(f(x, y, z)=2(x+y+z) / 3,0

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.
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