PROBABILITY THEORY:INDEPENDENT RANDOM VARIABLES AND RANDOM SAMPLES

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INDEPENDENT RANDOM VARIABLES AND RANDOM SAMPLES

The concept of independent events has already been defined; that is, E1 and E2 are inde- pendent events if, and only if,

INTRODUCTION TO OPERATIONS RESEARCH-0588

Thus, the independence of the random variables X1 and X2 implies that the joint CDF factors into the product of the CDF’s of the individual random variables. Furthermore, it is easily shown that if (X1,X2) is a discrete bivariate random variable, then X1 and X2 are independent random variables if, and only if, PX X (k, l) = PX (k)PX (l); in other words, P{X1 =

INTRODUCTION TO OPERATIONS RESEARCH-0589

for all s ant t. Thus, if X1, X2 are to be independent random variables, the joint density (or probability) function must factor into the product of the marginal density functions of the random variables. Using this result, it is easily seen that if X1, X2 are independent random variables, then the covariance of X1, X2 must be zero. Hence, the results on the variance of linear combinations of random variables given in Sec. 24.11 can be simpli- fied when the random variables are independent; that is,

INTRODUCTION TO OPERATIONS RESEARCH-0589

In other words, if X1 and X2 are independent, a knowledge of the outcome of one, say, X2, gives no information about the probability distribution of the other, say, X1. It was noted in the example in Sec. 24.10 on the time of first arrivals that the conditional den- sity of the arrival time of the first customer on the second day, given that the first cus- tomer on the first day arrived at time s, was equal to the marginal density of the arrival time of the first customer on the second day. Hence, X1 and X2 were independent random variables. In the example of the demand for a product during two consecutive months with the probabilities given in Sec. 24.9, it was seen in Sec. 24.10 that

INTRODUCTION TO OPERATIONS RESEARCH-0590Hence, the demands during each month were dependent (not independent) random variables.

The definition of independent random variables generally does not lend itself to de- termine whether or not random variables are independent in a probabilistic sense by look- ing at their outcomes. Instead, by analyzing the physical situation the experimenter usu- ally is able to make a judgment about whether the random variables are independent by ascertaining if the outcome of one will affect the probability distribution of the other.

The definition of independent random variables is easily extended to three or more random variables. For example, if the joint CDF of the n-dimensional random variable

INTRODUCTION TO OPERATIONS RESEARCH-0591

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