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.


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