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By Prof. F.P. Kelly

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N as n ! 1 9 a strong form of larger =1 This is NOT the same as the weak form. What does this mean? 2 determines Sn n n = 1 2 ::: as a sequence of real numbers. Hence it either tends to P ! ) ! as n ! 1 or it doesn’t. =1 30 CHAPTER 4. INEQUALITIES Chapter 5 Generating Functions In this chapter, assume that X is a random variable taking values in the range 0 Let pr = P(X = r) r = 0 1 2 : : : 1 2 : : :. 1. f) of the random variable X,or of the distribution pr = 0 1 2 : : : , is X X p(z ) = E z X = z r P(X = r) = pr z r 1 1 r=0 r=0 This p(z ) is a polynomial or a power series.

Note This holds for discrete random variables and is useful as a general way of finding the expectation whether the random variable is discrete or continuous. If X takes values in the set 0 1 : : : ] Theorem states E X] = 1 X n=0 P(X n) and a direct proof follows 1 X n=0 P(X n) = = = 1 X 1 X n=0 m=0 I m n]P(X = m) 1 X 1 X m=0 n=0 1 X m=0 ! 5. Let X be a continuous random variable with pdf f (x) and let h(x) be a continuous real-valued function. Then provided Z1 ;1 jh(x)j f (x)dx E h(x)] = 1 Z1 ;1 h(x)f (x)dx CHAPTER 6.

As n ! 1 or it doesn’t. =1 30 CHAPTER 4. INEQUALITIES Chapter 5 Generating Functions In this chapter, assume that X is a random variable taking values in the range 0 Let pr = P(X = r) r = 0 1 2 : : : 1 2 : : :. 1. f) of the random variable X,or of the distribution pr = 0 1 2 : : : , is X X p(z ) = E z X = z r P(X = r) = pr z r 1 1 r=0 r=0 This p(z ) is a polynomial or a power series. If a power series then it is convergent for jz j 1 by comparison with a geometric series. X jp(z )j r pr jz jr X r pr = 1 Example.

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