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Chapter 2: Problem 3

Let \(X_{1}, X_{2}, X_{3}\), and \(X_{4}\) be four independent random variables, each with pdf \(f(x)=3(1-x)^{2}, 0y)=P\left(X_{i}>y, i=1, \ldots, 4\right)\).

Short Answer

Expert verified
The cumulative distribution function (cdf) of \( Y \) is \( F_Y(y) = 1 - (1-y)^{12} \), and the probability density function (pdf) of \( Y \), is \( f_Y(y) = 36y(1-y)^{10} \), for \( y \in (0,1) \), and is zero elsewhere.

Step by step solution

01

Calculate the probability

The first step in our solution is to find \( P(Y>y) = P\left(X_{i}>y, i=1, \ldots, 4\right) \). For independent variables, the joint probability is the product of their individual probabilities, i.e., \( P(Y>y) = [P(X_{1}>y)P(X_{2}>y)P(X_{3}>y)P(X_{4}>y)] \). The function f(x) is nonzero in the range (0,1), so each of \( P(X_{i}>y) \) equates to \( \int_y^1{f(x)dx} \), for \( y \in (0,1) \). Evaluate this integral and compute the joint probability.
02

Compute the cdf

The cumulative distribution function (cdf) of \( Y \) is found as \( F_Y(y) = 1 - P(Y>y) \). Use the result from step 1 to find the cdf.
03

Derive the pdf

Finally, the pdf of a random variable is the derivative of its cdf. Thus, to find the pdf of \( Y \), denote \( f_Y(y) \), compute the derivative of \( F_Y(y) \) with respect to \( y \).

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Most popular questions from this chapter

Let 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?

Let \(X_{1}\) and \(X_{2}\) be two independent random variables so that the variances of \(X_{1}\) and \(X_{2}\) are \(\sigma_{1}^{2}=k\) and \(\sigma_{2}^{2}=2\), respectively. Given that the variance of \(Y=3 X_{2}-X_{1}\) is 25, find \(k\)

Let \(X_{1}, X_{2}\) be two random variables with joint \(\operatorname{pmf} p\left(x_{1}, x_{2}\right)=(1 / 2)^{x_{1}+x_{2}}\), for \(1 \leq x_{i}<\infty, i=1,2\), where \(x_{1}\) and \(x_{2}\) are integers, zero elsewhere. Determine the joint mgf of \(X_{1}, X_{2} .\) Show that \(M\left(t_{1}, t_{2}\right)=M\left(t_{1}, 0\right) M\left(0, t_{2}\right)\).

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

If the independent variables \(X_{1}\) and \(X_{2}\) have means \(\mu_{1}, \mu_{2}\) and variances \(\sigma_{1}^{2}, \sigma_{2}^{2}\), respectively, show that the mean and variance of the product \(Y=X_{1} X_{2}\) are \(\mu_{1} \mu_{2}\) and \(\sigma_{1}^{2} \sigma_{2}^{2}+\mu_{1}^{2} \sigma_{2}^{2}+\mu_{2}^{2} \sigma_{1}^{2}\), respectively.
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