By Howard G. Tucker and Ralph P. Boas (Auth.)

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8. Prove: If X, F, Z are three random variables such that FX,Y,Z(X, y, z) = Fx(x)FY (y)Fz{z) for all real x, y, z, then X, F, Z are independent. 1 Univariate Discrete Distributions Distributions A random variable X will be called discrete, and its distribution function Fx(x) will be called discrete, if the range of X is finite or denumerable. This means that there is a finite or denumerable set of real numbers X\, Xi, - - •, xny• • • such that En [ X = xn] = 12, where the union is taken over a finite set of integers in case the range of X is finite and over all positive integers in case the range of X is denumerable.

Repeated use of Theorems 1, 2, and 3 allows us to make the following sequence of statements: 2 X — F are random 2 (i) X + F and variables, 2 ( X — F )2 are random variables, and (ii) ( X + F ) and (iii) { ( X + F ) - ( X - F ) } / 4 is a random variable. Sec. 3] COMBINATIONS O F RANDOM VARIABLES 35 But this last random variable obviously equals X F , and thus XY is a random variable. If X and Y are random variables, then X/Y is a function which assigns to every co £ 12 the number X ( c o ) / F ( c o ) , provided F(co) ^ 0.

In the example given at the end of this section, compute P[X ^ 3]. 5 Multivariate Discrete Densities The joint discrete density function of n discrete random variables X2, - • -, Xn is defined by Xh Sec. 5] MULTIVARIATE DISCRETE DENSITIES fxi Xnfri, 51 "',Xn) The purpose of this section is to derive two important joint discrete density functions: the multivariate hypergeometric distribution and the multi nomial distribution. But first we must introduce the notion of conditional discrete densities.