12.1 - Poisson Distributions

12.1 - Poisson Distributions

Situation

Let the discrete random variable $$X$$ denote the number of times an event occurs in an interval of time (or space). Then $$X$$ may be a Poisson random variable with $$x=0, 1, 2, \ldots$$

Examples 12-1

1. Let $$X$$ equal the number of typos on a printed page. (This is an example of an interval of space — the space being the printed page.)
2. Let $$X$$ equal the number of cars passing through the intersection of Allen Street and College Avenue in one minute. (This is an example of an interval of time — the time being one minute.)
3. Let $$X$$ equal the number of Alaskan salmon caught in a squid driftnet. (This is again an example of an interval of space — the space being the squid driftnet.)
4. Let $$X$$ equal the number of customers at an ATM in 10-minute intervals.
5. Let $$X$$ equal the number of students arriving during office hours.
Poisson Random Variable

If $$X$$ is a Poisson random variable, then the probability mass function is:

$$f(x)=\dfrac{e^{-\lambda} \lambda^x}{x!}$$

for $$x=0, 1, 2, \ldots$$ and $$\lambda>0$$, where $$\lambda$$ will be shown later to be both the mean and the variance of $$X$$.

Recall that the mathematical constant e is the unique real number such that the value of the derivative (slope of the tangent line) of the function $$f(x)=e^x$$ at the point $$x=0$$ is equal to 1. It turns out that the constant is irrational, but to five decimal places, it equals:

$$\mathbf{e} = 2.71828$$

Also, note that there are (theoretically) an infinite number of possible Poisson distributions. Any specific Poisson distribution depends on the parameter $$\lambda$$.

"Derivation" of the p.m.f.

Let $$X$$ denote the number of events in a given continuous interval. Then $$X$$ follows an approximate Poisson process with parameter $$\lambda>0$$ if:

1. The number of events occurring in non-overlapping intervals are independent.
2. The probability of exactly one event in a short interval of length $$h=\frac{1}{n}$$ is approximately $$\lambda h = \lambda \left(\frac{1}{n}\right)=\frac{\lambda}{n}$$.
3. The probability of exactly two or more events in a short interval is essentially zero.

With these conditions in place, here's how the derivation of the p.m.f. of the Poisson distribution goes:

Now, let's make the intervals even smaller. That is, take the limit as $$n$$ approaches infinity $$n\rightarrow \infty$$ for fixed $$x$$. Doing so, we get:

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