Exponential & Gamma Laws | Probability Theory by E. Parzen

It has already been seen that the geometric and negative binomial probability laws arise in response to the following question: through how many trials need one wait in order to achieve the th success in a sequence of independent repeated Bernoulli trials in which the probability of success at each trial is ? In the same way, exponential and gamma probability laws arise in response to the question: how long a time need one wait if one is observing a sequence of events occurring in time in accordance with a Poisson probability law at the rate of events per unit time in order to observe the th occurrence of the event?

Example 4A. How long will a toll collector at a toll station at which automobiles arrive at the mean rate automobiles per minute have to wait before he collects the th toll for any integer ?

We now show that the waiting time to the th event in a series of events happening in accordance with a Poisson probability law at the rate of events per unit of time (or space) obeys a gamma probability law with parameter and ; consequently, it has probability density function

In particular, the waiting time to the first event obeys the exponential probability law with parameter (or equivalently, the gamma probability law with parameters and ) with probability density function

To prove (4.1), first find the distribution function of the time of occurrence of the th event. For , let denote the probability that the time of occurrence of the th event will be less than or equal to . Then represents the probability that the time of occurrence of the th event will be greater than . Equivalently, is the probability that the number of events occurring in the time from 0 to is less than ; consequently,

By differentiating (4.3) with respect to , one obtains (4.1).

Example 4B. Consider a baby who cries at random times at a mean rate of six distinct times per hour. If his parents respond only to every second time, what is the probability that ten or more minutes will elapse between two responses of the parents to the baby?

Solution

From the assumptions given (which may not be entirely realistic) the length in hours of the time interval between two responses obeys a gamma probability law with parameters and , Consequently,

in which the integral has been evaluated by using (4.3). If the parents responded only to every third cry of the baby, then

More generally, if the parents responded only to every th cry of the baby, then

The exponential and gamma probability laws are of great importance in applied probability theory, since recent studies have indicated that in addition to describing the lengths of waiting times they also describe such numerical valued random phenomena as the life of an electron tube, the time intervals between successive breakdowns of an electronic system, the time intervals between accidents, such as explosions in mines, and so on.

The exponential probability law may be characterized in a manner that illuminates its applicability as a law of waiting times or as a law of time to failure. Let be the observed waiting time (or time to failure). By definition, obeys an exponential probability law with parameter if and only if for every

It then follows that for any positive numbers and

In words, (4.7) says that, given an item of equipment that has served or more time units, its conditional probability of serving or more time units is the same as its original probability, when first put into service of serving or more time units. Another way of expressing (4.7) is to say that if the time to failure of a piece of equipment obeys an exponential probability law then the equipment is not subject to wear or to fatigue.

The converse is also true, as we now show. If the time to failure of an item of equipment obeys (4.7), then it obeys an exponential probability law. More precisely, let be the distribution function of the time to failure and assume that for for , and

Then necessarily, for some constant ,

If we define , then the foregoing assertion follows from a more general theorem.

Theorem. If a function satisfies the functional equation

and is bounded in the interval 0 to 1,

for some constant , then the function is given by

Proof

Suppose that (4.12) were not true. Then the function would not vanish identically in . Let be a point such that . Now it is clear that satisfies the functional equation in (4.10). Therefore, , and, for any integer . Consequently, . We now show that this cannot be true, since the function satisfies the inequality for all , in which is the constant given in (4.11). To prove this, note that . Since satisfies the functional equation in (4.10) it follows that, for any integer and for . Thus is a function that is periodic, with period 1. By (4.11), satisfies the inequality for . Being periodic with period 1, it therefore satisfies this inequality for all . The proof of the theorem is now complete.

For references to the history of the foregoing theorem, and a generalization, the reader may consult G. S. Young, “The Linear Functional Equation”, American Mathematical Monthly , Vol. 65 (1958), pp. 37–38.

Exercises

4.1. Consider a radar set of a type whose failure law is exponential. If radar sets of this type have a failure rate hours, find a length of time such that the probability is 0.99 that a set will operate satisfactorily for a time greater than .

Answer

hours.

4.2. The lifetime in hours of a radio tube of a certain type obeys an exponential law with parameter (i) , (ii) . A company producing these tubes wishes to guarantee them a certain lifetime. For how many hours should the tube be guaranteed to function, to achieve a probability of 0.95 that it will function at least the number of hours guaranteed?

4.3. Describe the probability law of the following random phenomenon: the number of times a fair die is tossed until an even number appears (i) for the first time, (ii) for the second time, (iii) for the third time.

Answer

obeys a negative binomial probability law with parameters and (i) , (ii) , (iii) .

4.4. A fair coin is tossed until heads appears for the first time. What is the probability that 3 tails will appear in the series of tosses?

4.5. The customers of a certain newsboy arrive in accordance with a Poisson probability law at a rate of 1 customer per minute. What is the probability that 5 or more minutes have elapsed since (i) his last customer arrived, (ii) his next to last customer arrived?

Answer

(i) 0.0067; (ii) 0.0404.

4.6. Suppose that a certain digital computer, which operates 24 hours a day, suffers breakdowns at the rate of 0.25 per hour. We observe that the computer has performed satisfactorily for 2 hours. What is the probability that the machine will not fail within the next 2 hours?

4.7. Assume that the probability of failure of a ball bearing at any revolution is constant and equal to . What is the probability that the ball bearing will fail on or before the th revolution? If , how many revolutions will be reached before of such ball bearings fail? More precisely, find so that , where is the number of revolutions to failure.

Answer

4.8. A lepidopterist wishes to estimate the frequency with which an unusual form of a certain species of butterfly occurs in a particular district. He catches individual specimens of the species until he has obtained exactly 5 butterflies of the form desired. Suppose that the total number of butterflies caught is equal to 25. Find the probability that 25 butterflies would have to be caught in order to obtain 5 of a desired form, if the relative frequency of occurrence of butterflies of the desired form is given by (i) , (ii) .

4.9. Consider a shop at which customers arrive at random at a rate of 30 per hour. What fraction of the time intervals between successive arrivals will be (i) longer than 2 minutes, (ii) shorter than 4 minutes, (iii) between 1 and 3 minutes.

Answer

(i) 0.368; (ii) 0.865; (iii) 0.383.

The Exponential and Gamma Probability Laws

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