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Multivariate normal distribution

Multivariate normal distribution

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LESSON 2 MULTIVARIATE NORMAL DISTRIBUTION

2.0 Introduction

In this Lesson we shall consider non singular multivariate normal distribution including

marginal and conditional distributions of subsets.

2.1 Objectives

By the end of this Lesson, you shall be able to:

1. Write down the probability density function of the multivariate normal random

vector

2. Show that the function given in 1. above is indeed a probability density function

by verifying the two conditions of a pdf

3. Obtain the moment generating function(mgf ) of the multivariate normal density.

4. Infer the marginal distribution of the components of the multivariate normal

vector

2.2 . The probability density and moment generating function.

We recall that a univariate normal random variable X with mean µ and variance

has the

density function

1 (x −µ)

f x( ) = exp{− 2σ

}

2.1

σ π2

and we denote this by X N~ (µ,

)

In the multivariate case, we say that p-variate random vector X follows the multivaraiate

(or p-variate) normal distribution(MVN) with mean vectorµ and variance-covariance

MVN(µ, Σ) or X ~ N

(µ, Σ)if the joint density( matrix Σ, denoted by X ~

pdf ) is given

by:

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−

(x −µ)} 2.2 f ( )x

= exp{− (x −µ) Σ

(2π)

Whereµ is a p×1 vector of constants and Σ is a p× p symmetric positive definite

matrix of constants. For p =1 equation 2.2 reduces to equation 2.1.

Write 2.2 for the case p=2 andσ

= ρσσ

ij i j

. What do you call this

distribution?

If X

, X

, ..., X

are independent random variables such that X

~ N(µ

,σ

) , then the joint pdf

of X

, X

, ..., X

is simply the product of the marginal density functions so that

p 2

1 1 ⎛ xi −µi ⎞ }

f x x( 1, 2 , ..., xp ) = (2 )π σp2 ∏pexp{− 2 ∑i=1 ⎜⎝ σi ⎟⎠

−

(x −µ)} 2.3

= exp{− (x −µ) Σ

(2π)

Where µ′ = (µ µ

, ...,µ

)

and Σ = diag(σ

,σ

, ...,σ

) . Thus the joint distribution of the

random variables X

, X

,..., X

is a special case of the MVN pdf in 2.2.

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LESSON 2 MULTIVARIATE NORMAL DISTRIBUTION 2.0 Introduction In this Lesson we shall consider non singular multivariate normal distribution including marginal and conditional distributions of subsets. 2.1 Objectives By the end of this Lesson, you shall be able to: 1. Write down the probability density function of the multivariate normal random vector 2. Show that the function given in 1. above is indeed a probability density function by verifying the two conditions of a pdf 3. Obtain the moment generating function(mgf ) of the multivariate normal density. 4. Infer the marginal distribution of the components of the multivariate normal vector 2.2 . The probability density and moment generating function. We recall that a univariate normal random variable X with mean µ and variance σ2 has the density function 1 (x −µ)2 f x( ) = exp{− 2σ2 } 2.1 σ π2 and we denote this by X N~ (µ,σ2) In the multivariate case, we say that p-variate random vector X follows the multivaraiate (or p-variate) normal distribution(MVN) with mean vectorµ and variance-covariance matrix Σ, denoted by pdf ) is given X ~ MVN(µ, Σ) or X ~ Np(µ, Σ)if the joint density( by: 1 (x −µ)} ' −1 1 1 1 exp{− (x −µ) Σ (2π) 2 Σ 2 2 p = 2.2 f ( )x Whereµ is a p×1 vector of constants and Σ is a p× p symmetric positive definite matrix of constants. For p =1 equation 2.2 reduces to equation 2.1. Write 2.2 for the case p=2 andσij = ρσσij i j . What do you call this distr ...
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