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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Tags: Binomial Distribution, Math, Monday Math, Poisson, Poisson Distribution, Poisson Process, Probability, Shot Noise, Statistics

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