Statistics 510: Notes 7



Statistics 510: Notes 15

Reading: Section 5.5-5.6.1, 9.1

I. The Poisson Process (Section 9.1)

The Poisson process is a model for events occurring randomly in time. Examples include: the arrival of clients at a bank, the occurrence of serious earthquakes, telephone calls to a call center, the occurrence of power outages, urgent calls to an emergency center, etc.

Events over a time period t occur according to a Poisson process if

(a) the probability of an event occurring in a given small time period [pic]is approximately proportion to [pic]

(b) the probability of two or more events occurring in a given small time period [pic]is much smaller than [pic]

(c) the number of events occurring in two non-overlapping time periods are independent.

Another way of describing the Poisson process is

(1) The events occur one by one.

(2) The number of events occurring during any two nonoverlapping time intervals are independent of one another.

(3) The number of arrivals during any given time interval has a Poisson distribution of which the expected value is proportional to the duration of the interval.

The parameter [pic]is called the rate or arrival intensity of the Poisson process.

II. Exponential Random Variables

Example (Example 4.7B): Suppose that earthquakes in the western portion of the United States occur according to a Poisson process with [pic]and with 1 week as the unit of time, i.e.,

Find the probability distribution of the time, starting from now, until the next earthquake. What is the probability that the time is greater than two weeks?

Exponential Random Variables: A random variable with CDF [pic]is called an exponential random variable with parameter [pic].

The probability density function of an exponential random variable is for [pic]

[pic]

and 0 for [pic]

The time until the first event in a Poisson process with rate [pic] is an exponential random variable with parameter [pic].

Example 1: Suppose customers arrive at a bank according to a Poisson process with arrival intensity 2 per minute. What is the probability that starting at 1 p.m., the first customer arrives within two minutes?

Mean and variance of exponential random variable:

We will show that [pic].

For n>0, we have

[pic]

Integrating by parts ([pic]) yields

[pic]

Thus,

[pic]

Memorylessness of exponential random variable:

Let X be an exponential random variable with parameter [pic]. We have for all [pic]

[pic]

In other words,

[pic] (1.1)

If we think of X as the lifetime of some electrical device, equation (1.1) states the probability that the device survives for at least [pic]hours given that it has survived t hours is the same as the initial probability that it survives for at least s hours. In other words, if the device is alive at age t, the distribution of the remaining amount of time that it survives is the same as the original lifetime distribution (that is, it is as if the instrument does not remember that it has already been in use for a time t). This is called the memorylessness property of the exponential distribution.

Example 2: Is the exponential distribution a good model for the distribution of human lifetimes?

Example 3: Consider a post office that is staffed by two clerks. Suppose that when Mr. Smith enters the post office, he discovers that Ms. Jones is being served by one of the clerks and Mr. Brown by the other. Suppose also that Mr. Smith is told that his service will begin as soon as either Jones or Brown leaves. If the amount of time that a clerk spends with a customer is exponentially distributed with parameter [pic], what is the probability that, of the three customers, Mr. Smith is the last to leave the post office?

III. Gamma distribution:

Suppose events occur according to a Poisson process with arrival intensity [pic].

Suppose we start observing the process at some time (which we denote by time 0). The time until the first event occurs has an exponential ([pic]) distribution.

Let X denote the time until the first [pic]events occur. Let W denote the number of occurrences of the event in the interval [pic]. Then W is a Poisson random variable with parameter [pic]. The cdf of X can be obtained using W as follows:

[pic]

Therefore,

[pic]

What we have just derived is a special case of the gamma family of probability distributions.

The gamma family can be generalized to cases in which [pic]is positive but not necessarily an integer. To do this, we replace [pic]with a continuous function of (nonnegative) [pic], [pic], the latter reducing to [pic]when [pic]is a positive integer.

For any real number [pic], the gamma function (of [pic]) is given by

[pic].

Let X be a random variable such that

[pic]

Then X is said to have a gamma distribution with parameters [pic] and [pic].

The mean and variance of the gamma distribution are

[pic].

The gamma family of distributions is a flexible family of probability distributions for modeling nonnegative valued random variables. The following plot shows the pdfs for some gamma distributions.

[pic]

Example 1 continued: Suppose customers arrive at a bank according to a Poisson process with arrival intensity 2 per minute. What is the probability that starting at 1 p.m., the first two customers have arrived within three minutes? What is the expected value of the amount of time it takes for two customers to arrive?

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