## Monday Math 31

The binomial distribution, the discrete probability distribution of obtaining n successes out of N Bernoulli trials, each with probability p of sucess, is
.
We see that the probabilities sum to unity due to the binomial theorem:
.
Now, consider the expected number of successes:

Now, note that the n=0 term is zero, and may be dropped:

Shifting the index by defining k=n-1 and M=N-1, we have
.

Now, suppose we rewrite it in terms of the expected number of successes  in place of p:
.

Let us consider what happens if we take the limit as the sample size , while holding λ fixed:


Now, in that first fraction, the numerator has n terms:
,
and so as N increases without bound, each of these terms approaches unity. The next fraction,  is independent of N.
In the last term, , the base of the exponent approaches unity, while the exponent is fixed, so that term in turn approaches unity.
Lastly, we recall that
,
and so we see that the penultimate term fits this form, and so
.

This distribution is known as the Poisson distribution, which is the distribution for a large number of trials of a Poisson process, and which models a number of real-world processes, such as radioactive decay, or shot noise.