PROBABILITY THEORY:BIVARIATE PROBABILITY DISTRIBUTION

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BIVARIATE PROBABILITY DISTRIBUTION

Thus far the discussion has been concerned with the probability distribution of a single random variable, e.g., the demand for a product during the first month or the demand for a product during the second month. In an experiment that measures the demand during the first 2 months, it may well be important to look at the probability distribution of the vector random variable (X1, X2), the demand during the first month, and the demand during the second month, respectively,

INTRODUCTION TO OPERATIONS RESEARCH-0577

dom variable X1 taken on values less than or equal to b1, and X2 takes on values less than or equal to b2. Then P{EX , X2} denotes the probability of this event. In the above example of the demand for a product during the first 2 months, suppose that the sample space n consists of the set of all possible points w, where w represents a pair of nonnegative integer values (x1,x2). Assume that x1 and x2 are bounded by 99. Thus, there are (100)2w points in n. Suppose further that each point w has associated with it a probability equal to 1/(100)2, except for the points w = (0,0) and w = (99,99). The probability associated with the event {0,0} will be 1.5/(100)2, that is, P{0,0} = 1.5/(100)2, and the probability associated with the event {99,99} will be 0.5/(100)2; that is, P{99,99} = 0.5/(100)2. Thus, if there is interest in the “bivariate” random variable (X1, X2), the demand during the first and second months, respectively, then the event

INTRODUCTION TO OPERATIONS RESEARCH-0577

A similar calculation can be made for any value of b1 and b2.

For any given bivariate random variable (X1, X2), P{X1 ::: b1, X2 ::: b2} is denoted by

INTRODUCTION TO OPERATIONS RESEARCH-0578

INTRODUCTION TO OPERATIONS RESEARCH-0579

(which is analogous to what was done for a univariate random variable). This causes no difficulty, even for bivariate random variables having one or more components that can- not take on negative values or are restricted to other regions. In this case, fX X (s, t) can be defined to be zero over the inadmissible part of the plane. In fact, the only requirements for a function to be a bivariate density function are that

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