each is in Software
EXAMPLE 4.3 each is Scan QRCode In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Making QR In None Using Barcode creation for Software Control to generate, create QR Code image in Software applications. 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 Denso QR Bar Code Reader In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. QR Code ISO/IEC18004 Encoder In Visual C#.NET Using Barcode maker for Visual Studio .NET Control to generate, create Quick Response Code image in VS .NET applications. P(X1
Painting Denso QR Bar Code In .NET Framework Using Barcode generation for ASP.NET Control to generate, create Quick Response Code image in ASP.NET applications. Quick Response Code Drawer In VS .NET Using Barcode maker for VS .NET Control to generate, create QR Code 2d barcode image in VS .NET applications. 2, X2
Paint QRCode In VB.NET Using Barcode printer for Visual Studio .NET Control to generate, create QRCode image in .NET framework applications. Bar Code Generator In None Using Barcode creator for Software Control to generate, create bar code image in Software applications. 2, c, X6
Print Bar Code In None Using Barcode printer for Software Control to generate, create bar code image in Software applications. Code 128 Code Set A Generator In None Using Barcode maker for Software Control to generate, create ANSI/AIM Code 128 image in Software applications. The expected number of times that A1, A2, . . . , Ak will occur in n trials are np1, np2, . . . , npk respectively, i.e., Draw Code 3/9 In None Using Barcode encoder for Software Control to generate, create Code39 image in Software applications. EAN128 Encoder In None Using Barcode creation for Software Control to generate, create USS128 image in Software applications. E(X1) Identcode Generator In None Using Barcode encoder for Software Control to generate, create Identcode image in Software applications. Paint Bar Code In ObjectiveC Using Barcode creation for iPhone Control to generate, create barcode image in iPhone applications. np1, Encoding Code 39 In Java Using Barcode generator for Java Control to generate, create Code 39 Full ASCII image in Java applications. Barcode Printer In .NET Using Barcode drawer for Reporting Service Control to generate, create barcode image in Reporting Service applications. E(X2) Creating 1D Barcode In Java Using Barcode printer for Java Control to generate, create 1D image in Java applications. Barcode Generation In Java Using Barcode printer for Java Control to generate, create barcode image in Java applications. np2, Scan Barcode In .NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET applications. EAN13 Generator In Java Using Barcode maker for BIRT reports Control to generate, create UPC  13 image in BIRT applications. ..., 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 41, 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)

