- O.H Probability and Markov Chains – MATH 2561/2571 E09 1 Generating functions Even quite straightforward counting problems can lead to

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Unformatted text preview: 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 ≥ , 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 ≥ . Example 1.2 (Binomial Theorems) For integer n , we have (1 + s ) n = n summationdisplay k =0 parenleftbigg n k parenrightbigg s k , n ≥ (1.2) and (1- s ) − n = ∞ summationdisplay k =0 parenleftbigg n + k- 1 k parenrightbigg s k , n ≥ , | 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 ≥ is the convolution of ( a k ) k ≥ and ( b m ) m ≥ (write c = a ⋆ b ), if c n = n summationdisplay k =0 a k b n − k , n ≥ , (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 ≥ have generating function G a ( s ) . Find the generating function of the sequence c n def = a + a 1 + ··· + a n . Solution. Since ( c n ) n ≥ is the convolution of ( a k ) k ≥ and ( b m ) m ≥ 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 O.H. Probability and Markov Chains – MATH 2561/2571 E09 Definition 1.6 Suppose that X is a discrete random variable taking values in { , 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 ≥ E ( X k ) k !...
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This note was uploaded on 05/12/2010 for the course APPLIED ST 2010 taught by Professor Various during the Spring '10 term at Universidad Nacional Agraria La Molina.

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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

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