The next in the line of common discrete probability distributions is called the Poisson distribution.
When we have the rate of success and we’re interested in the probability of witnessing
some number of successes given that rate, that number of successes follows a Poisson
distribution. Here, “rate” implies “occurrences per some unit.”
For example, in queueing theory, the hourly arrivals to a machine in a production process,
or the monthly occurrence of breakdowns of a piece of equipment. In quality control,
the rate of defects per 100 cars, or the number of scratches per slab of granite. In radar
engineering, images appear on a radar screen at a particular rate. In civil infrastructure,
potholes appear with some occurrence per mile. Again, the rate is “occurrence per some
unit,” and the random variable counts the number of occurrences. The independence of
occurrences is assumed: seeing one occurrence doesn’t affect the likelihood of another
occurrence. Also, the random variable must be discrete, as we’re counting the number
of times we witness an occurrence of something. But the rate need not be discrete, as the
rate is an average. That is, we could see 2.6 defects per 100 cars on average. But the
actual observation of defects per 100 cars must be discrete.
The probability distribution function for the probability that random variable Y takes
on a value y is lambda to the y times e to the minus lambda divided by y factorial. Here,
lambda is the rate of occurrence, and e is the common Euler’s number found in mathematics,
equal to 2.718. This is the probability of y occurrences in a given interval.
Say you’re a machine operator at Vandelay Industries, a company that manufactures latex.
Say that latex stock arrives at your work station at an average of 3.2 batches every
10 minutes. What is the probability of having more than 7 batches in a 10-minute interval?
Let’s define random variable X to be the arrivals of latex stock batches in 10 minutes.
The rate of arrival is lambda = 3.2 batches/10 minutes. Note that the interval in the rate
has to match the interval in the random variable. So the probability that Y equals 7 is 3.2
to the 7 times e to the negative 3.2 divided by 7 factorial. This probability is 0.0278.
What is the probability of exactly 10 batches arriving during a 20 minute interval? Let’s
define random variable Y as the number of latex batches arriving during a 20 minute
interval, and the rate of occurrence would then be 6.4 batches per 20 minutes. If 3.2
batches arrive per 10 minutes, we would naturally expect double that to occur in 20 minutes.
So the probability that y equals 10, takes on the value 10 is 6.4 to the 10 times e to
the negative 6.4 all divided by 10 factorial.
The calculation of the expected value of a Poisson random variable requires a Taylor
series expansion. As it turns out, the expected value of Poisson random variable Y is simply
lambda. And that shouldn’t be too surprising: how many arrivals would we expect to see per
10 minutes? The average arrival rate per 10 minutes. And, the variance is lambda squared,
therefore the standard deviation is lambda.
A couple things to note about the Poisson distribution. It differs from the binomial
and hypergeometric distributions in that it doesn’t have an upperbound. Naturally, the
lowerbound is zero, like the other distributions, as we can’t have a negative count of occurrences.
But here, the count of observations extends indefinitely, though with very small probabilities.
So, generating a probability distribution wouldn’t be complete: one would likely stop
calculating probabilities of particular realizations when those probabilities got sufficiently
small. Also, there is a relationship between the Poisson distribution and a continuous
distribution that we’ll discuss later: given the rate of occurrence, lambda, the Poisson
distribution counts occurrences while the exponential distribution describes the interval
between occurrences, where the interval is continuous, most often describing often time.
For example, if customers arrive with rate lambda equals 5.2 customers per hour, the
number of customers arriving in an hour follows a Poisson distribution, while the hours between
arrivals follows an exponential distribution.