lecture13

# lecture13 - Kunal Mehta 15913699 Branching Processes...

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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

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1 1 0 p 0 With mean = 1 With mean > 1 Not possible Next random sums
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lecture13 - Kunal Mehta 15913699 Branching Processes...

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