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LESSON3: DISTRIBUTION OF LINEAR FUNCTIONS OF MVN, MARGINAL

AND CONDITIONAL DISTRIBUTIONS

3.0 Introduction

In this Lesson we shall consider distribution of linear function of multivariate normal

variables, marginal and conditional distributions of subsets of the multivariate normal

vector.

3.1 Objectives

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

• Derive the distribution of a linear function of components of the multivariate

normal vector.

• Use matrix approach and the result in 1 above to obtain the marginal distribution of

a subset of the multivariate normal vector.

• Write down the conditional distribution of a subset of the multivariate normal

vector given another subset of the vector.

3.2 Distribution of linear functions of MVN

Let X ~ N

(µ, Σ) or X ~ MVN(µ, Σ)and let Y = c

′

X =

∑cX

i i

where c

′

= (c c

1 2

, , ..., c

)

i=1

and c

are real numbers not all equal to zero. We wish to find the pdf of Y .

The mgf M

( )t of the distribution ofY is given by

( )t = E e[

]= E e[

tc X

′

]

Now,

exp{t′µ+ t′Σt} exists for all real values of t . Thus we

M( )t = E e[

t X

′

can replace t′ by

tc′ and obtain

′µ+ 1c′Σct

} 3.1

( )t = exp{tc

Thus by uniqueness of mgf the random variable Y has a univariate normal distribution i.e.

Y N~ (c′µ, c′Σc)

What do you say about the distribution of a

linear function of components of a multivariate

vector? Write down the pdf of Y .

Next let Y

= A

r p

X , where A is a matrix of constants and X ~ N

(µ, Σ).

( )t = E e[

t Y′

]

= E e[

′

], where t

a r×1vector

= E e[

s′ X

], where s′ = t A′

=exp{s

′µ+ 1 s′Σs}

=exp{s

′Aµ+ 1 s′A AΣ ′s}

′E A( X) + 1t′ var(AX) }t 3.2

=exp{t

Thus by uniqueness of mgf Y ~ N A A A

( µ,Σ ′) .

What do you say about the distribution of a linear function of a

multivariate vector? Write down the pdf of Y .

3.3. Marginal and conditional distribution of subsets

3.3.1 Marginal distribution of a subset

Consider a random vector X

(X X

, ..., X

)′ partitioned into sub vectors

= (X

, X

, ..., X

)′ and X

= (X X

r+1

r+2

, ..., X

)′

Note that partitioning can be done arbitrarily so that X

is a r×1 vector and X

is a q×1

vector where r q+ = p .

Suppose

E(X

) =µ

=(µ

,µ

, ...,µ

)′

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LESSON3: DISTRIBUTION OF LINEAR FUNCTIONS OF MVN, MARGINAL AND CONDITIONAL DISTRIBUTIONS 3.0 Introduction In this Lesson we shall consider distribution of linear function of multivariate normal variables, marginal and conditional distributions of subsets of the multivariate normal vector. 3.1 Objectives By the end of this Lesson, you shall be able to: • Derive the distribution of a linear function of components of the multivariate normal vector. • Use matrix approach and the result in 1 above to obtain the marginal distribution of a subset of the multivariate normal vector. • Write down the conditional distribution of a subset of the multivariate normal vector given another subset of the vector. 3.2 Distribution of linear functions of MVN p Let X ~ Np(µ, Σ) or X ~ MVN(µ, Σ)and let Y = c X = ∑cXi ′ ′ i where c = (c c1 2, , ..., cp) i=1 and ci are real numbers not all equal to zero. We wish to find the pdf of Y . The mgf MY ( )t of the distribution ofY is given by MY ( )t = E e[ tY ]= E e[ tc X′ ] Now, M( )t = E e[ t X′ exp{t′µ+ ]= t′Σt} exists for all real values of t . Thus we can replace t′ by tc′ and obtain ′µ+ 1c′Σct2} 3.1 MY ( )t = exp{tc 2 Thus by uniqueness of mgf the random variable Y has a univariate normal distribution i.e. Y N~ (c′µ, c′Σc) What do you say about the distribution of a linear function of components of a multivariate vector? Write down the pdf of Y . 1 Next let Y r×1 = Ar p× X , where A is a ...
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