2-The Poisson random variable
A random variable X that takes on one of the values 0, 1, 2, . . . is said
to be a Poisson random variable with parameter λ if, for some λ > 0,
This equation defines a probability mass function, since
The Poisson random variable has a tremendous range of applications in diverse areas because
it may be used as an approximation for a binomial random variable with parameters (n, p)
when n is large and p is small enough so that np is of moderate size. To see this,
suppose that X is a binomial random variable with parameters (n, p), and let λ = np. Then
Now, for n large and λ moderate,
Hence, for n large and λ moderate,
Some examples of random variables that generally obey the Poisson probability law are as follows:
1. The number of misprints on a page (or a group of pages) of a book
2. The number of people in a community who survive to age 100
3. The number of wrong telephone numbers that are dialed in a day
4. The number of packages of dog biscuits sold in a particular store each day
5. The number of customers entering a post office on a given day
6. The number of vacancies occurring during a year in the federal judicial system
7. The number of α-particles discharged in a fixed period of time from some radioactive material
Suppose that the number of typographical errors on a single page of this book has a Poisson distribution
with parameter λ = 
.
Calculate the probability that there is at least one error on this page. Solution.
Letting X denote the number of errors on this page, we have
π(π ≥1)=1−π(π =0)=1−
≈0.393
Suppose that the probability that an item produced by a certain machine will
be defective is 0.1. Find the probability that a sample of 10 items will contain at most 1 defective item.
The desired probability is
whereas the Poisson approximation yields the value
Consider an experiment that consists of counting the number of α particles
given off in a 1-second interval by 1 gram of radioactive material.
If the gram of radioactive material as consisting of a large number n of atoms,
each of which has probability of 3.2/n of disintegrating and sending off an α particle during
the second considered, what is a good approximation to the probability that no more than 2 α particles will appear?
The gram of radioactive material consisting of a large number n of atoms,
each of which has probability of 3.2/n of disintegrating and sending off an α particle during the second considered,
then we see that, to a very close approximation, the number of α particles given off will be
a Poisson random variable with parameter λ = 3.2. Hence, the desired probability is
Before computing the expected value and variance of the Poisson random variable
with parameter λ, recall that this random variable approximates a binomial random variable
with parameters n and p when n is large, p is small, and λ = np. Since such a binomial random variable
has expected value np = λ and variance np(1 − p) = λ(1 − p) L λ (since p is small),
it would seem that both the expected value and the variance of a Poisson random variable would equal its parameter λ.
We now verify this result:
Thus, the expected value of a Poisson random variable X is indeed equal to its parameter λ.
To determine its variance, we first compute
where the final equality follows because the first sum is the expected value of a Poisson random variable
with parameter λ and the second is the sum of the probabilities of this random variable.
Therefore, since we have shown that E[X] = λ, we obtain
Hence, the expected value and variance of a Poisson random variable are both equal to its parameter λ.
*Properties of the cumulative distribution function
Recall that, for the distribution function F of X, F(b) denotes
the probability that the random variable X takes on a value that is less than or equal to b.
Following are some properties of the cumulative distribution function (c.d.f.) F:
1. F is a nondecreasing function; that is, if a < b, then F(a)≤F(b).
2. 
πΉ(π)=1.
3. πΉ(π)=0.
4. F is right continuous. That is, for any b and any decreasing sequence bn, π≥1, that converges to b,
Property 1 follows, because, for a < b, the event {X ≤a} is contained in the event {X ≤b} and so cannot have a larger probability.
Properties 2, 3, and 4 all follow from the continuity property of probabilities All
probability questions about X can be answered in terms of the c.d.f., F. For example,
π(π <π ≤π)=πΉ(π)−πΉ(π) πππ πππ π <π
This equation can best be seen to hold if we write the event {X ≤ b} as the union of the mutually exclusive events
{X ≤a} and {a < X ≤b}. That is,
{π ≤π}={π ≤π}∪{π <π≤π}
so
π(π ≤π)=π(π ≤π)+π(π <π ≤π)
If we want to compute the probability that X is strictly less than b,
we can again apply the continuity property to obtain
Note that P{X < b} does not necessarily equal F(b), since F(b) also includes the probability that X equals b.
The distribution function of the random variable X is given by
A graph of F(x) is presented as
Compute (a) P{X < 3}, (b) P{X = 1}, (c) P{X > 12}, and (d) P{2 < X ≤ 4}.
