A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 6: Q. 6.42 (page 274)

The joint density of X and Y is
$f (x, y) = c (x^{2} - y^{2}) e^{- x} 0 鈮� x < 鈭�, - x 鈮� y 鈮� x$
Find the conditional distribution of Y, given X = x.

Short Answer

Expert verified

For $t 鈮� x :$

$P (Y 鈮� t 鈭� X = x) = \frac{3}{2 x^{3}} (x^{2} (t + x) - \frac{t^{3}}{3} + \frac{x^{3}}{3})$

For $t < - x :$

$P (Y 鈮� t 鈭� X = x) = 0$

For $t > x :$

$P (Y 鈮� t 鈭� X = x) = 1$

Step by step solution

Given information

Joint density of X and Y

$f (x, y) = c (x^{2} - y^{2}) e^{- x} 0 鈮� x < 鈭�, - x 鈮� y 鈮� x$

also $X = x$

Formula to be used :

$f (y 鈭� x) = \frac{f (x, y)}{f (x)}$

Explanation

$鈭� f (x, y) d y = {鈭�}_{- x}^{x} c (x^{2} - y^{2}) e^{- x} d x = \frac{2 c}{3} x^{3} e^{- x}, 0 < x < 鈭� T h u s, f (y 鈭� x) = \frac{3}{2 x^{3}} (x^{2} - y^{2}), - x 鈮� y 鈮� x$

By integration,

For $t 鈮� x,$

localid="1647240753725" role="math" $P (Y 鈮� t 鈭� X = x) = {鈭�}_{- x}^{t} f (x, y) d y = \frac{3}{2 x^{3}} (x^{2} (t + x) - \frac{t^{3}}{3} + \frac{x^{3}}{3})$

Then for t < -x,

$P (Y 鈮� t 鈭� X = x) = 0$

and for t > x,

$P (Y 鈮� t 鈭� X = x) = 1$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

The joint density of X and Y is
$f (x, y) = c (x^{2} - y^{2}) e^{- x} 0 鈮� x < 鈭�, - x 鈮� y 鈮� x$
Find the conditional distribution of Y, given X = x.

Short Answer

Step by step solution

Given information

Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Logic and Functions

Mechanics Maths

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

91影视

The joint density of X and Y isf(x,y)=cx2-y2e-x0鈮�x<鈭�,-x鈮�y鈮�xFind the conditional distribution of Y, given X = x.

Short Answer

Step by step solution

Given information

Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Logic and Functions

Mechanics Maths

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

The joint density of X and Y is
$f (x, y) = c (x^{2} - y^{2}) e^{- x} 0 鈮� x < 鈭�, - x 鈮� y 鈮� x$
Find the conditional distribution of Y, given X = x.