lecture13 - Kunal Mehta 15913699 Branching Processes...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Kunal Mehta 15913699 Branching Processes Lecture 13 Probability Generating Function : " ( s ) = generic notation for PGF ( s ) = s n p n n = 0 # $ where ( p 0 , p 1 , p 2 ,...) is the probability distribution of some r.v. x : P ( X = n ) = p n E ( s x ) = ( s ) To make connection between x and φ , write: ( s ) = x ( s ) Fact : E ( x ) = d ds ( s ) s = 1 # *Note ( s ) may only be defined for s with s $ 1 E x ( x # 1) [ ] = d 2 ds 2 ( s ) s = 1 # Look at sums : Take x and y , independent random variables Consider : x + y ( s ) = E ( s x + y ) = E ( s x s y ) = E ( s x ) E ( s y ) # by independence = x ( s ) y ( s ) Generalize to equation S n = X 1 + X 2 + ... + X n with X i independent : S n ( s ) = ( x ( s )) n Application : S n = sum of die rolls Find P ( S n = k ) x ( s ) = 1 6 ( s + s 2 + ... + s 6 ) Compute S n ( s ) = 1 6 ( s + s 2 + ... + s 6 ) # $ % ( n = polynomial in s ) use computer algebra
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
1 1 0 p 0 With mean = 1 With mean > 1 Not possible Next random sums
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

lecture13 - Kunal Mehta 15913699 Branching Processes...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online