Binomial Distribution

https://en.wikipedia.org/wiki/Binomial_distribution

Random Variable X has a binomial distribution with parameters (n,p)

https://www.s-cool.co.uk/a-level/maths/probability-distributions/revise-it/the-binomial-distribution

Binomial distribution is defined as, where q = 1-p :

• In summary, a sum of independent Bernoulli random variables is a binomial random variable.

Binomial Expectation:

E[x] = np

• proof:

E[x] = np

Var[x] = np(1-p)

Var[x] = np(1-p)

Moment generating function:

M(t) = (pe^t + (1-p))^n

Bernoulli distribution properties:

• If the random variables X1, X2, …, Xn are n random variables from a Bernoulli trial with parameter p, then X = X1 + X2 + … + Xn has a Binomial distribution with parameters (n,p)
• If X1, X2, …, Xk are independent random variables, and Xi has the binomial distribution with parameters ni and p ( for example X1 ~B(n1,p) . . . , ~B(nk,p)), then X = X1 + X2 + … + Xk has binomial distribution with parameters n = n1 + · · · + nk and p, ~B(n1 + · · · + nk,p).