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0809GeneratingFunctions2H - O.H Probability and Markov...

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O.H. Probability and Markov Chains – MATH 2561/2571 E09 1 Generating functions Even quite straightforward counting problems can lead to laborious and lengthy calculations. These are greatly simplified by using generating functions. 3 Definition 1.1 Given a collection of real numbers ( a k ) k 0 , the function G ( s ) = G a ( s ) def = summationdisplay k =0 a k s k (1.1) is called the generating function of ( a k ) k 0 . Example 1.2 (Binomial Theorems) For integer n , we have (1 + s ) n = n summationdisplay k =0 parenleftbigg n k parenrightbigg s k , n 0 (1.2) and (1 - s ) n = summationdisplay k =0 parenleftbigg n + k - 1 k parenrightbigg s k , n 0 , | s | < 1 . (1.3) Solution. The first formula follows immediately by straightforward combinatorics; for the second, use the n th power of 1 1 - s = X k =0 s k , | s | < 1 . Definition 1.3 We say that a sequence ( c n ) n 0 is the convolution of ( a k ) k 0 and ( b m ) m 0 (write c = a ⋆ b ), if c n = n summationdisplay k =0 a k b n k , n 0 , (1.4) Theorem 1.4 (Convolution) If c = a⋆b , then the generating functions G c ( s ) , G a ( s ) , and G b ( s ) satisfy G c ( s ) = G a ( s ) G b ( s ) . Proof. An easy exercise. square Example 1.5 Let a sequence ( a k ) k 0 have generating function G a ( s ) . Find the generating function of the sequence c n def = a 0 + a 1 + · · · + a n . Solution. Since ( c n ) n 0 is the convolution of ( a k ) k 0 and ( b m ) m 0 with b m 1, the theorem and (1.3) imply that G c ( s ) = G a ( s ) G b ( s ) G a ( s ) / (1 - s ). 3 introduced by de Moivre and Euler in the early eighteenth century. 1
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O.H. Probability and Markov Chains – MATH 2561/2571 E09 Definition 1.6 Suppose that X is a discrete random variable taking values in { 0 , 1 , 2 , . . . } . The ( probability ) generating function of the random variable X is the generating function G X ( s ) of its probability mass function, G ( s ) G X ( s ) def = E ( s X ) = summationdisplay k =0 s k P ( X = k ) . (1.5) Remark 1.6.1 Observe that the moment generating function M X ( s ) def = E ( e sX ) = X k 0 E ( X k ) k ! s k of a random variable X , is the generating function of the sequence E ( X k ) /k !. Example 1.7 If X Poi ( λ ) , ie P ( X = k ) = λ k ( k !) 1 e λ , then G X ( s ) = summationdisplay k 0 s k P ( X = k ) = summationdisplay k 0 ( λs ) k k !
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