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In words, we say that the standardized random variable (X
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Let X be a discrete random variable that can take on the values 0, 1, 2, . . . such that the probability function of X is given by f (x) P(X x) lxe x!
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where is a given positive constant. This distribution is called the Poisson distribution (after S. D. Poisson, who discovered it in the early part of the nineteenth century), and a random variable having this distribution is said to be Poisson distributed. The values of f (x) in (13) can be obtained by using Appendix G, which gives values of e for various values of .
Some Properties of the Poisson Distribution
Some important properties of the Poisson distribution are listed in Table 4-3.
Table 4-3
Mean Variance Standard deviation Coefficient of skewness Coefficient of kurtosis Moment generating function Characteristic function s a3 a4 M(t) f(v) 3
!l 1> !l (1>l) el(et el(eiv
1) 1)
Relation Between the Binomial and Poisson Distributions
In the binomial distribution (1), if n is large while the probability p of occurrence of an event is close to zero, so that q 1 p is close to 1, the event is called a rare event. In practice we shall consider an event as rare if the number of trials is at least 50 (n 50) while np is less than 5. For such cases the binomial distribution is very closely approximated by the Poisson distribution (13) with np. This is to be expected on comparing Tables 4-1 and 4-3, since by placing np, q < 1, and p < 0 in Table 4-1, we get the results in Table 4-3.
CHAPTER 4 Special Probability Distributions
Relation Between the Poisson and Normal Distributions
Since there is a relation between the binomial and normal distributions and between the binomial and Poisson distributions, we would expect that there should also be a relation between the Poisson and normal distributions. This is in fact the case. We can show that if X is the Poisson random variable of (13) and (X l)> !l is the corresponding standardized random variable, then
lim P aa S
l !l
b 1 3a e !2p
u2>2
(14) l)> !l is asymptotically
i.e., the Poisson distribution approaches the normal distribution as l S ` or (X normal.
The Central Limit Theorem
The similarity between (12) and (14) naturally leads us to ask whether there are any other distributions besides the binomial and Poisson that have the normal distribution as the limiting case. The following remarkable theorem reveals that actually a large class of distributions have this property. Theorem 4-2 (Central Limit Theorem) Let X1, X2, . . . , Xn be independent random variables that are identically distributed (i.e., all have the same probability function in the discrete case or density function in the continuous case) and have finite mean and variance 2. Then if Sn X1 X2 . . . Xn (n l, 2 . . .),
lim Paa S
b 1 e !2p 3a
u2>2
(15)
that is, the random variable (Sn to Sn, is asymptotically normal.
nm)>s!n, which is the standardized variable corresponding
The theorem is also true under more general conditions; for example, it holds when X1, X2, . . . , Xn are independent random variables with the same mean and the same variance but not necessarily identically distributed.
The Multinomial Distribution
Suppose that events A1, A2, . . . , Ak are mutually exclusive, and can occur with respective probabilities p1, p2, . . . , pk where p1 p2 c pk 1. If X1, X2, . . . , Xk are the random variables respectively giving the number c X n, then of times that A1, A2, . . . , Ak occur in a total of n trials, so that X1 X2 k P(X1 n1, X2 n2, c, Xk nk) n pn1pn2 c pnk k n1!n2! c nk! 1 k (16)
where n1 n2 c nk n, is the joint probability function for the random variables X1, c, Xk. This distribution, which is a generalization of the binomial distribution, is called the multinomial distribution c p )n. since (16) is the general term in the multinomial expansion of ( p1 p2 k
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