Transformations - University of Arizona

Hierarchical Models

Convolution

Chapter 4

Examples of Mass Functions and Densities

Transformations

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Hierarchical Models

Convolution

Outline

Convolution

Hierarchical Models

Bernoulli Censored Poisson

Normal Random Variables

Bayes Formula Revisited

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Hierarchical Models

Convolution

Convolution

Let

s = x1 + x2 ,

y = x1 .

Then,

x2 = s ? y

x1 = y ,



J(s, y ) = det

0 1

1 ?1



= ?1

This yields

fS,Y (s, y ) = fX1 ,X2 (y , s ? y ).

The marginal distribution for S = X1 + X2 can be found by taking an integral

Z ¡Þ

fX1 +X2 (s) =

fX1 ,X2 (x1 , s ? x1 ) dx1 .

?¡Þ

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Hierarchical Models

Convolution

Gamma Distribution

If X1 and X2 are independent with sum S, then

Z ¡Þ

fS (s) =

fX1 (x1 )fX2 (s ? x1 ) dx1 .

?¡Þ

This is called the convolution and is often written fX1 +X2 = fX1 ? fX2 . For Xi ¡« ¦£(¦Ái , ¦Â)

fS (s)

=

=

=

=

s

¦Â ¦Á1 ¦Á1 ?1 ?¦Âx1 ¦Â ¦Á2

x1

e

(s ? x1 )¦Á2 ?1 e ?¦Â(s?x1 ) dx1 .

¦£(¦Á2 )

0 ¦£(¦Á1 )

Z s

¦Â ¦Á1 +¦Á2

?¦Âs

e

x1¦Á1 ?1 (s ? x1 )¦Á2 ?1 dx1 .

¦£(¦Á1 )¦£(¦Á2 )

0

Z 1

¦Â ¦Á1 +¦Á2

s ¦Á1 +¦Á2 ?1 e ?¦Âs

u ¦Á1 ?1 (1 ? u)¦Á2 ?1 du, u = x1 /s.

¦£(¦Á1 )¦£(¦Á2 )

0

¦Â ¦Á1 +¦Á2

¦£(¦Á1 )¦£(¦Á2 )

¦Â ¦Á1 +¦Á2

s ¦Á1 +¦Á2 ?1 e ?¦Âs

=

s ¦Á1 +¦Á2 ?1 e ?¦Âs

¦£(¦Á1 )¦£(¦Á2 )

¦£(¦Á1 + ¦Á2 )

¦£(¦Á1 + ¦Á2 )

Z

and X1 + X2 ¡« ¦£(¦Á1 + ¦Á2 , ¦Â)

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Hierarchical Models

Convolution

Gamma Distribution

Continuing,

fS,X1 (s, x1 )

fX ,X (x1 , s ? x1 )

= 1 2

.

fS (s)

fS (s)

For the example on the sum of gamma random variables

fX1 |S (x1 |s) =

fX1 |S (x1 |s) =

=

=

¦Â ¦Á1 +¦Á2

?¦Âs x ¦Á1 ?1 (s ? x )¦Á2 ?1

1

1

¦£(¦Á1 )¦£(¦Á2 ) e

,

¦Á

+¦Á

1

2

¦Â

¦Á1 +¦Á2 ?1 e ?¦Âs

s

¦£(¦Á1 +¦Á2 )

¦£(¦Á1 +¦Á2 ) ¦Á1 ?1

(s ? x1 )¦Á2 ?1

¦£(¦Á1 )¦£(¦Á2 ) x1

s ¦Á1 +¦Á2 ?1

0 ¡Ü x1 ¡Ü s

¦£(¦Á1 + ¦Á2 )  x1 ¦Á1 ?1 

x1 ¦Á2 ?1 1

1?

¦£(¦Á1 )¦£(¦Á2 ) s

s

s

Thus, X1 |S ¡« S ¡¤ Beta(¦Á1 , ¦Á2 ) and E [X1 |S] = ¦Á1 S/(¦Á1 + ¦Á2 )

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