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Statistics / Mathematical Statistics with Applications 7 / Chapter 5 / Problem 128E

Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer

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Let Y1 and Y2 have a bivariate normal distribution.a Show

Chapter 5, Problem 128E

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QUESTION:

Problem 128E

Let Y1 and Y2 have a bivariate normal distribution.

a Show that the marginal distribution of Y1 is normal with mean μ1 and variance σ 21 .

b What is the marginal distribution of Y2?

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QUESTION:

Problem 128E

Let Y1 and Y2 have a bivariate normal distribution.

a Show that the marginal distribution of Y1 is normal with mean μ1 and variance σ 21 .

b What is the marginal distribution of Y2?

ANSWER:

Solution:

Step 1 of 2:

Let ${Y}_{1}$ and ${Y}_{2}$ have bivariate normal distribution

The claim is to show the marginal distribution of ${Y}_{1}$ is normal with mean ${\mu{}}_{1}$ and variance ${\sigma{}}_{1}^{2}$

The marginal distribution function of ${Y}_{1}$ is

${f}_{1}({y}_{1})$ = $\int\limits_{-\infty{}}^{\infty{}}f({y}_{1,}\ {y}_{2})\$ d ${y}_{2}$

Make change for variable $\mu{}\ =({y}_{1}-{\mu{}}_{1})$ and y = $\frac{{y}_{2}-{\mu{}}_{2}}{{\sigma{}}_{2}}$

Then,

${f}_{1}({y}_{1})$ = $\frac{1}{2\Pi{}{\sigma{}}_{1}\sqrt{1\ -\ {\rho{}}^{2}}}$ $\int\limits_{-\infty{}}^{\infty{}}exp(\ -\ \frac{1}{2(1\ -\ {\rho{}}^{2})\ }({\mu{}}^{2}+{v}^{2}-2\rho{}\mu{}v)\$ dv

To evaluate the above equation

${\mu{}}^{2}+{v}^{2}-2\rho{}\mu{}v$ = ${(v-\rho{}\mu{})}^{2}$ + ${\mu{}}^{2}$ (1 - ${\rho{}}^{2}$ )

So that

${f}_{1}({y}_{1})$ = $\frac{1}{2\Pi{}{\sigma{}}_{1}\sqrt{1\ -\ {\rho{}}^{2}}}$ ${e}^{-\frac{{\mu{}}^{2}}{2}}$ $\int\limits_{-\infty{}}^{\infty{}}exp(\ -\ \frac{1}{2(1\ -\ {\rho{}}^{2})\ }{(v-\rho{}\mu{})}^{2}\$ dv

The integral is that of a normal density with mean $\rho{}\mu{}$ variance 1 - ${\rho{}}^{2}$

Therefore, ${f}_{1}({y}_{1})$ = $\frac{1}{2\Pi{}{\sigma{}}_{1}\sqrt{1\ -\ {\rho{}}^{2}}}$ ${e}^{-\frac{{({y}_{1}-{\mu{}}_{1})}^{2}}{2{\sigma{}}_{1}^{2}}}$ - $\infty{}\ <{y}_{1\ }<\infty{}$

Hence,

The marginal distribution function of ${Y}_{1}$ is ${f}_{1}({y}_{1})$ = $\frac{1}{2\Pi{}{\sigma{}}_{1}\sqrt{1\ -\ {\rho{}}^{2}}}$ ${e}^{-\frac{{({y}_{1}-{\mu{}}_{1})}^{2}}{2{\sigma{}}_{1}^{2}}}$ - $\infty{}\ <{y}_{1\ }<\infty{}$

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Solution Found!

Let Y1 and Y2 have a bivariate normal distribution.a Show

Questions & Answers

Study Tools You Might Need

Chapter: 6 Problem: 6

Chapter: 10 Problem: 65

Chapter: 28 Problem: 28.40

Chapter: 8 Problem: 16

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Chapter: 33 Problem: 10

Chapter: 23 Problem: 23.17

Chapter: 8 Problem: 55

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