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If a fair die is to be tossed 12 times, the probability of getting 1, 2, 3, 4, 5 and 6 points exactly twice 12! 1 2 1 2 1 2 1 2 1 2 1 2 a b a b a b a b a b a b 2!2!2!2!2!2! 6 6 6 6 6 6 1925 559,872
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The expected number of times that A1, A2, . . . , Ak will occur in n trials are np1, np2, . . . , npk respectively, i.e.,
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E(Xk)
(17)
The Hypergeometric Distribution
Suppose that a box contains b blue marbles and r red marbles. Let us perform n trials of an experiment in which a marble is chosen at random, its color is observed, and then the marble is put back in the box. This type of experiment is often referred to as sampling with replacement. In such a case, if X is the random variable denoting
CHAPTER 4 Special Probability Distributions
the number of blue marbles chosen (successes) in n trials, then using the binomial distribution (1) we see that the probability of exactly x successes is P(X x) n b xr n x , a b x (b r)n x 0, 1, c, n (18)
since p b>(b r), q 1 p r>(b r). If we modify the above so that sampling is without replacement, i.e., the marbles are not replaced after being chosen, then b r a ba b x n x a b n r b
max (0, n min (n, b)
r), c,
(19)
This is the hypergeometric distribution. The mean and variance for this distribution are m nb b r , s2 (b nbr(b r n) r)2 (b r 1) (20)
If we let the total number of blue and red marbles be N, while the proportions of blue and red marbles are p and q 1 p, respectively, then p b b r b , N q r b r r N or b Np, r Nq (21)
so that (19) and (20) become, respectively, a P(X x) Nq Np ba b x n x N a b n s2 npq(N n) N 1 n a b p xqn x
(22)
(23)
Note that as N S ` (or N is large compared with n), (22) reduces to (18), which can be written P(X and (23) reduces to np,
(24)
(25)
in agreement with the first two entries in Table 4-1, page 109. The results are just what we would expect, since for large N, sampling without replacement is practically identical to sampling with replacement.
The Uniform Distribution
A random variable X is said to be uniformly distributed in a f (x) e 1>(b 0 a) x b if its density function is (26) a x b otherwise
and the distribution is called a uniform distribution. The distribution function is given by F(x) P(X x)
u (x
a)>(b
x x x
a b b
(27)
CHAPTER 4 Special Probability Distributions
The mean and variance are, respectively, m 1 (a 2 b), s2 1 (b 12 a)2 (28)
The Cauchy Distribution
A random variable X is said to be Cauchy distributed, or to have the Cauchy distribution, if the density function of X is a f (x) (29) a 0, ` x ` p(x2 a2) This density function is symmetrical about x 0 so that its median is zero. However, the mean, variance, and higher moments do not exist. Similarly, the moment generating function does not exist. However, the characteristic function does exist and is given by ( ) e
(30)
The Gamma Distribution
A random variable X is said to have the gamma distribution, or to be gamma distributed, if the density function is xa 1e x>b x 0 a (a, b 0) (31) f (x) u b (a) x 0 0 where ( ) is the gamma function (see Appendix A). The mean and variance are given by ,
(32)
The moment generating function and characteristic function are given, respectively, by M(t) (1 t) , ( ) (1 i ) (33)
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