Q.7.4 The joint density function ofXan... [FREE SOLUTION]

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A First Course in Probability

Sheldon M. Ross

$Math Studyset 魅影直播 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 7: Q.7.4 (page 355)

The joint density function of $X$ and $Y$ is given by
$f (x, y) = \frac{1}{y} e^{- (y + x / y)}, x > 0, y > 0$
Find $E [X], E [Y]$ and show that $C o v (X, Y) = 1$

Short Answer

Expert verified

The value of $E [X] = 1$

The value of $E [Y] = 1$

The value of $Cov (X, Y) = 1$

Step by step solution

Given Information

The density of $X$ and $Y$ is $f (x, y) = \frac{1}{y} e^{- (y + x / y)}, x > 0, y > 0$

$E [X] = ?$

$E [Y] = ?$

$Cov (X, Y) = ?$

Explanation

Calculate the value of $E [X],$

$E [X] = {鈭�}_{0}^{鈭�} {鈭�}_{0}^{鈭�} x 路 \frac{1}{y} e^{- (y + \frac{x}{y})} d y d x$

$= {鈭�}_{0}^{鈭�} e^{- y} [{鈭�}_{0}^{鈭�} \frac{x}{y} e^{- \frac{x}{y}} d x] d y$

$= {鈭�}_{0}^{鈭�} y e^{- y} d y$

$= {y \frac{e^{- y}}{- 1}|}_{0}^{鈭�} - {鈭�}_{0}^{鈭�} \frac{e^{- y}}{- 1} d y$

$= 0 + 1$

$= 1$

Explanation

Calculate the value of $E [Y],$

$E [Y] = {鈭�}_{0}^{鈭�} {鈭�}_{0}^{鈭�} y 路 \frac{1}{y} e^{- (y + \frac{x}{y})} d x d y$

$= {鈭�}_{0}^{鈭�} {鈭�}_{0}^{鈭�} e^{- y} e^{- \frac{x}{y}} d x d y$

$= {鈭�}_{0}^{鈭�} e^{- y} [{鈭�}_{0}^{鈭�} e^{- \frac{x}{y}} d x] d y$

$= {鈭�}_{0}^{鈭�} e^{- y} y d y$

$= 1$

Explanation

Calculate the value of $Cov (X, Y),$

$E (X Y) = {鈭�}_{0}^{鈭�} {鈭�}_{0}^{鈭�} x y 路 \frac{1}{y} e^{- (y + \frac{x}{y})} d x d y$

$= {鈭�}_{0}^{鈭�} e^{- y} [{鈭�}_{0}^{鈭�} x e^{\frac{- x}{y}} d x] d y$

$= {鈭�}_{0}^{鈭�} e^{- y} [{\frac{x e^{\frac{- x}{y}}}{\frac{- 1}{y}}|}_{0}^{鈭�} - {鈭�}_{0}^{鈭�} \frac{e^{\frac{- x}{y}}}{\frac{- 1}{y}} d x] d y$

$= {鈭�}_{0}^{鈭�} e^{- y} [0 + y^{2}] d y$

$= {鈭�}_{0}^{鈭�} y^{2} e^{- y} d y$

$= - {y^{2} e^{- y}|}_{0}^{鈭�} + {鈭�}_{0}^{鈭�} 2 y e^{- y} d y$

$= 0 + 2 = 2$

Final Answer

Hence $Cov (X, Y) = E (X Y) - E (X) E (Y) = 2 - 1.1$

$= 2 - 1$

$= 1$

So, the value of $C o v (X, Y) = 1$

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魅影直播

The joint density function of $X$ and $Y$ is given by
$f (x, y) = \frac{1}{y} e^{- (y + x / y)}, x > 0, y > 0$
Find $E [X], E [Y]$ and show that $C o v (X, Y) = 1$

Short Answer

Step by step solution

Given Information

Explanation

Explanation

Explanation

Final Answer

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魅影直播

The joint density function ofXandYis given byf(x,y)=1ye-(y+x/y),x>0,y>0Find E[X],E[Y]and show that Cov(X,Y)=1

Short Answer

Step by step solution

Given Information

Explanation

Explanation

Explanation

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Mechanics Maths

Theoretical and Mathematical Physics

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

The joint density function of $X$ and $Y$ is given by
$f (x, y) = \frac{1}{y} e^{- (y + x / y)}, x > 0, y > 0$
Find $E [X], E [Y]$ and show that $C o v (X, Y) = 1$