Chapter 6: Q. 6.41 (page 274)

The joint density function of X and Y is given by
$f (x, y) = x e^{- x (y + 1)} x > 0, y > 0$
(a) Find the conditional density of X, given Y = y, and that of Y, given X = x.
(b) Find the density function of Z = XY.

Short Answer

Expert verified

The conditional density of X is $x (y + 1)^{2} e^{- x (y + 1)} a n d x e^{- x y}$ when Y = y, and X = x respectively.

Step by step solution

Given information (part a)

$f (x, y) = x e^{- x (y + 1)} x > 0, y > 0$

Explanation

the conditional density of X and Y = y is

$f_{x y} (x ∣ y) = \frac{f (x, y)}{f_{y} (y)} = \frac{x e^{- x (y + 1)}}{\frac{1}{(y + 1)^{2}}} = x (y + 1)^{2} e^{- x (y + 1)}$

the conditional density of X when X = x is,

$f_{x y} (y ∣ x) = \frac{f (x, y)}{f_{y} (x)} = \frac{x e^{- x (y + 1)}}{e^{- x}} = x e^{- x y}$

Given information (part b)

Density function $Z = X Y$

Explanation (part b)

The cumulative distributive function of the random variable $Z = X Y$ is,

$F_{z} (a) = P (Z \leq a) = P (X Y \leq a) = P (X > 0, Y \leq \frac{a}{x}) = \int_{0}^{\infty} \int_{0}^{a / x} f (x, y) d y d x = \int_{0}^{\infty} \int_{0}^{a / x} x e^{- x (y + 1)} d y d x = \int_{0}^{\infty} x e^{- x} \int_{0}^{a / x} e^{- x y} d y d x = \int e^{- x} {[\frac{e^{- x y}}{- x}]}_{0}^{a / x} d x = \int e^{- x} [e^{- x (a / x)} - 1] d x = \int e^{- x} [e^{- a} - 1] d x = (1 - e^{- a}) \int_{0}^{\infty} e^{- x} d x = (1 - e^{- a}) {[\frac{e^{- x}}{- 1}]}_{0}^{\infty} = 1 - e^{- a}$

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The joint density function of X and Y is given by
$f (x, y) = x e^{- x (y + 1)} x > 0, y > 0$
(a) Find the conditional density of X, given Y = y, and that of Y, given X = x.
(b) Find the density function of Z = XY.

Short Answer

Step by step solution

Given information (part a)

Explanation

Given information (part b)

Explanation (part b)

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The joint density function of X and Y is given byf(x,y)=xe-x(y+1)x>0,y>0(a) Find the conditional density of X, given Y = y, and that of Y, given X = x.(b) Find the density function of Z = XY.

Short Answer

Step by step solution

Given information (part a)

Explanation

Given information (part b)

Explanation (part b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

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Mechanics Maths

Study anywhere. Anytime. Across all devices.

The joint density function of X and Y is given by
$f (x, y) = x e^{- x (y + 1)} x > 0, y > 0$
(a) Find the conditional density of X, given Y = y, and that of Y, given X = x.
(b) Find the density function of Z = XY.