Answered: Suppose the random variables X and Y…

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Suppose the random variables X and Y have a pdf given by f (x, y) x+y on 0 Y). prob = | b. Find F(,을): ans =

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 19E

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Expert Solution

Introduction

Given that,

$X$ and $Y$ are two random variables

The joint pdf of them is given by ,

$f (x, y) = \{\begin{cases} x + y & 0 < x < 1, 0 < y < 1 \\ 0 & otherwise \end{cases}$

Step 1

(a) $P (X > Y)$ is to be obtained.

It can be written as $P (X > Y) = 1 - P (X < Y)$

Marginal pdf of $X$ is given by

$\begin{array}{rcl} \int_{0}^{1} (x + y) d y \\ = & \int_{0}^{1} x d y + \int_{0}^{1} y d y \\ = & x {(y)}_{0}^{1} + {(\frac{y^{2}}{2})}_{0}^{1} \\ = & x + \frac{1}{2} \end{array}$

Now,

$\begin{array}{rcl} P (X < Y) & = & \int_{0}^{y} x + \frac{1}{2} d x \\ = & {(\frac{x^{2}}{2})}_{0}^{y} + {(\frac{x}{2})}_{0}^{y} \\ = & \frac{y^{2}}{2} + \frac{y}{2} \end{array}$

Therefore, $P (X > Y) = 1 - \begin{array}{rcl} \frac{y^{2}}{2} \end{array} \begin{array}{rcl} + \end{array} \begin{array}{rcl} \frac{y}{2} \end{array}$

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