Lecture 22: Moment Generating Functions & Applications in Statistics - STAT 702/J702 by Br, Study notes of Statistics

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STAT 702/J702 B.Habing Univ. of S.C. (^1)

STAT 702/J

November 14 th, 2006

- Lecture 22-

Instructor: Brian Habing Department of Statistics Telephone: 803-777- E-mail: [email protected]

STAT 702/J702 B.Habing Univ. of S.C. (^2)

Today

• Last Time - Covariance
• Moment Generating Functions
• Applications!

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Covariance

Cov(X,Y )=E[(X -μX )(Y -μY )]

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Correlation

Cor(X,Y )= ( ) ( )

( , )

Var XVar Y

Cov XY

ρ XY =

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4.5 – Moment Generating Functions

The moment-generating function

(mgf) of X isM(t )=E(etX^ )

= ∑

x

M ( t ) etX^ p ( x )

= ∫

∞

−∞

M ( t ) etX^ f ( x ) dx

STAT 702/J702 B.Habing Univ. of S.C. (^6)

Why “moment generating”? Assume the mgf exists on some interval around 0…

= ∫

∞

−∞

M ( t ) etX^ f ( x ) dx

= ∫

∞

−∞

e f x dx

dt

d

M '( t ) tX^ ( )

STAT 702/J702 B.Habing Univ. of S.C. (^7)

Other properties: a) The m.g.f. uniquely determines the p.d.f.

b) If Y=a +bX then M Y (t )=eat^ MX (bt )

c) If X and Y are independent and

Z=X+Y then M Z(t )=M X (t ) M Y (t )

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Example 1) X~Uniform[0,1]

MX (t)=

MaX+b(t)=

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Example 2) Sum of Negative Binomials.

for ln( 1 ) [ 1 ( 1 ) ]

( )

( ) t p pe

pe M t t r

tr <− − − −

=

STAT 702/J702 B.Habing Univ. of S.C. (^10)

Application 3: Intelligent Searching and Sampling

a) Group Testing: A large numbern

of blood samples are to be tested for a relatively rare disease. Can we find all the infected samples in

fewer thann tests?

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Consider the case of splitting each of

n samples in half. Combine half of

each one is placed into a large combined pool.

Should this work better?

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Now consider that we divide them

samples intom groups of sizek

each…