Binomial Distribution

#probability #distribution #discrete_distribution

Gist

The binomial distribution is The number of times some outcome occurs out of n independent trials, each having the same probability of “success” p.

Mathematics

The PMF is :

P (X = k) = (\binom{n}{k}) p^{k} (1 - p)^{n - k}, k = 0, 1, \dots, n .

Binomial Coefficient

If n is the number of objects in a set and k is the subset. The binomial coefficient is

(\binom{n}{k}) = \frac{n!}{(n - k)! k!},

where we read this as " $n$ choose $k$ "

Properties

Relationship

You can have a normal approximation Normal Distribution to the binomial theorem (under specific conditions) through the
The distribution converges to a Poisson Distributions as n increases, p goes to infinity, and np approaches $λ$

Confidence interval

Interesting article about not using Wald: https://www.econometrics.blog/post/don-t-use-the-textbook-ci-for-a-proportion/

References

http://stratigrafia.org/8370/rtips/proportions.html