PROBABILITY THEORY:DISCRETE PROBABILITY DISTRIBUTIONS

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DISCRETE PROBABILITY DISTRIBUTIONS

It was pointed out in Sec. 24.2 that one is usually concerned with random variables and their associated probability distributions, and discrete random variables are those which take on a finite or countably infinite set of values. Furthermore, Sec. 24.3 indicates that the CDF for a random variable is given by

INTRODUCTION TO OPERATIONS RESEARCH-0552

INTRODUCTION TO OPERATIONS RESEARCH-0553

INTRODUCTION TO OPERATIONS RESEARCH-0554

Such a random variable is said to have a Bernoulli distribution. Thus, if a random variable takes on two values, say, 0 or 1, with probability 1 - p or p, respectively, a Bernoulli random variable is obtained. The upturned face of a flipped coin is such an example: If a head is denoted by assigning it the number 0 and a tail by assigning it a 1, and if the coin is “fair” (the probability that a head will appear is 1/2), the upturned face is a Bernoulli random variable with parameter p = 1/2. Another example of a Bernoulli random variable is the quality of an item. If a defective item is denoted by 1 and a nondefective item by 0, and if p represents the probability of an item being defective, and 1 - p represents the probability of an item being nondefective, then the “quality” of an item (defective or non- defective) is a Bernoulli random variable.

If X1, X2, . . . , Xn are independent1 Bernoulli random variables, each with parameter p, then it can be shown that the random variable

INTRODUCTION TO OPERATIONS RESEARCH-0555

1The concept of independent random variables is introduced in Sec. 24.12. For the present purpose, random variables can be considered independent if their outcomes do not affect the outcomes of the other random variables.

INTRODUCTION TO OPERATIONS RESEARCH-0556

The geometric distribution is useful in the following situation. Suppose an experiment is performed that leads to a sequence of independent1 Bernoulli random variables, each with parameter p; that is, P{X1 = 1} = p and P(X1 = 0) = 1 - p, for all i. The random variable X, which is the number of trials occurring until the first Bernoulli random variable takes on the value 1, has a geometric distribution with parameter p.

1The concept of independent random variables is introduced in Sec. 24.12. For now, random variables can be considered independent if their outcomes do not affect the outcomes of the other random variables.

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