Poisson Random Variables

Number of occurrences Let Y represent the number of occurrences of an event in an interval of size s. Here we may be referring to an interval of time, distance, space, etc. For example, we may be interested in the number of customers Y arriving during a given time interval. We call Y a Poisson random variable.

Poisson R. V. A random variable has a Poisson distribution with parameter l if its probability function is given by where y = 0, 1, 2, … We’ll see that l is the “average rate” at which the events occur. That is, E(Y) = l .

Queries If the number of database queries processed by a computer in a time interval is a Poisson random variable with an average of 6 queries per minute, find the probability that 4 queries occur in a one minute interval.

poissonpdf(l, y) poissoncdf(l, y) is also provided on the TI-83

Fewer Queries As before, for the Poisson random variable with an average of 6 queries per minute… find the probability there are less than 6 queries in a one minute interval:

Some PoissonVariables Number of incoming telephone calls to a switchboard within a given time interval; Number of errors (incorrect bits) received by a modem during a given time interval; Number of chocolate chips in one of Dr. Vestal’s chocolate chip cookies; Number of claims processed by a particular insurance company on a single day; Number of white blood cells in a drop of blood; Number of dead deer along a mile of highway.

Many short intervals To derive the Poisson probability distribution, think of the interval as being comprised of many, say n, very short successive intervals. x x x x x x x x x x s Suppose in each interval, either there is an occurrence or there’s not. “like Bernoulli trials” So Y = y occurrences is like y successes in n trials. Treat it like a binomial experiment, but let the time intervals get very short (i.e., let n get very large).

As n goes to infinity… Let l = np, the expected number of successes. Taking the limit 1 e-l

As n goes to infinity… With l = np held constant, as n gets large, we have found Consequently, we may use a Poisson probability to approximate binomial probabilities when n is large (and p is small). ( Suggests large n and l = np < 7.)

Compare Consider a binomial experiment with n = 200 and p = 0.03, so that l = np = 6. Determine the probability of 4 successes. Also, approximate the probability using Poisson distribution.

Poisson mean, variance If Y is a Poisson random variable with parameter l, the expected value and variance for Y are given by ( and the proof is too good to pass up)

Expected number of arrivals The expected value for a Poisson random variable: since first term is zero, start with y = 1 cancelling the common factor distributing out one l

Expected number of arrivals The expected value for a Poisson random variable: If y = 1, 2, 3,… then y – 1 = 0, 1, 2, … Let z = y – 1. =1 a sum of Poisson probabilities , as claimed.

Deriving Variance Deriving the variance for a Poisson random variable proceeds in a similar manner. As we’ve seen before, to get E(Y2), you first determine that E[Y(Y – 1)] = l2. (how?) And so, E(Y2) = E[Y(Y – 1)] + E(Y) = l2 + l. Finally, V(Y) = E(Y2) – [E(Y)]2 = (l2 + l) - l2 = l.