PROBABILITY THEORY:CONTINUOUS PROBABILITY DISTRIBUTIONS

CONTINUOUS PROBABILITY DISTRIBUTIONS

Section 24.2 defined continuous random variables as those random variables that take on a continuum of values. The CDF for a continuous random variable FX(b) can usually be written as

where fX(y) is known as the density function of the random variable X. From a notational standpoint, the subscript X is used to indicate the random variable that is under consideration. When there is no ambiguity, this subscript may be deleted, and fX(y) will be de- noted by f(y). It is evident that the CDF can be obtained if the density function is known. Furthermore, a knowledge of the density function enables one to calculate all sorts of probabilities, for example,

Note that strictly speaking the symbol P{a < X ::: b} relates to the probability that the outcome w of the experiment belongs to a particular event in the sample space, namely, that event such that X(w) is between a and b whenever w belongs to the event. However, the reference to the event will be suppressed, and the symbol P will be used to refer to the probability that X falls between a and b. It becomes evident from the previous ex- pression for P{a < X ::: b} that this probability can be evaluated by obtaining the area under the density function between a and b, as illustrated by the shaded area under the density function shown in Fig. 24.8. Finally, if the density function is known, it will be said that the probability distribution of the random variable is determined.

Naturally, the density function can be obtained from the CDF by using the relation

For a given value c, P{X = c} has not been defined in terms of the density function. However, because probability has been interpreted as an area under the density function, P{X = c} will be taken to be zero for all values of c. Having P{X = c} = 0 does not mean that the appropriate event E in the sample space (E contains those w such that X(w) = c) is an impossible event. Rather, the event E can occur, but it occurs with probability zero. Since X is a continuous random variable, it takes on a continuum of possible values, so that selecting correctly the actual outcome before experimentation would be rather star- tling. Nevertheless, some outcome is obtained, so that it is not unreasonable to assume that the preselected outcome has probability zero of occurring. It then follows from P{X = c} being equal to zero for all values c that for continuous random variables, and any a and b,

In defining the CDF for continuous random variables, it was implied that fX(y) was defined for values of y from minus infinity to plus infinity because

The exponential distribution has had widespread use in operations research. The time between customer arrivals, the length of time of telephone conversations, and the life of electronic components are often assumed to have an exponential distribution. Such an assumption has the important implication that the random variable does not “age.” For example, suppose that the life of a vacuum tube is assumed to have an ex- ponential distribution. If the tube has lasted 1,000 hours, the probability of lasting an additional 50 hours is the same as the probability of lasting an additional 50 hours, given that the tube has lasted 2,000 hours. In other words, a brand new tube is no “better” than one that has lasted 1,000 hours. This implication of the exponential distribution is quite important and is often overlooked in practice.

The Gamma Distribution

A continuous random variable whose density is given by

A graph of a typical gamma density function is given in Fig. 24.11.

A random variable having a gamma density is useful in its own right as a mathe- matical representation of physical phenomena, or it may arise as follows: Suppose a cus- tomer’s service time has an exponential distribution with parameter 8. The random vari- able T, the total time to service n (independent) customers, has a gamma distribution with parameters n and 8 (replacing a and {3, respectively); that is,

Note that when n = 1 (or a = 1) the gamma density becomes the density function of an exponential random variable. Thus, sums of independent, exponentially distributed ran- dom variables have a gamma distribution.

Another important distribution, the chi square, is related to the gamma distribution. If X is a random variable having a gamma distribution with parameters {3 = 1 and a = v/2 (v is a positive integer), then a new random variable Z = 2X is said to have a chi- square distribution with v degrees of freedom. The expression for the density function is given in Table 24.1 at the end of Sec. 24.8.

The Beta Distribution

A continuous random variable whose density function is given by

is known as a beta-distributed random variable. This density is a function of the two pa- rameters a and {3, both of which are positive constants. A graph of a typical beta density function is given in Fig. 24.12.

Beta distributions form a useful class of distributions when a random variable is re- stricted to the unit interval. In particular, when a = {3 = 1, the beta distribution is called the uniform distribution over the unit interval. Its density function is shown in Fig. 24.13, and it can be interpreted as having all the values between zero and 1 equally likely to oc- cur. The CDF for this random variable is given by

In this case -(b - JL)/u is positive, and FX(b) = a is thereby read from the table by entering it with -(b - JL)/u. Thus, suppose it is desired to evaluate the expression

P{2 < X ::: 18} = FX(18) - FX(2).

FX(18) has already been shown to be equal to 1 - 0.0228 = 0.9772. To find FX(2) it is first noted that (2 - 10)/4 = -2 is negative. Hence, Table A5.1 is entered with Ka = +2, and FX(2) = 0.0228 is obtained. Thus, FX(18) - FX(2) = 0.9772 - 0.0228 = 0.9544.

As indicated previously, the normal distribution is a very important one. In particular, it can be shown that if X1, X2, . . . , Xn are independent,1 normally distributed random 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.