Binomial Distribution

This actually comes from the binomial theorem.

We say that $X$ is a random variable has a binomial distribution with parameters $p$ and $n$ , and we write $X \sim bin (n, p)$ . If $X$ has a pmf of the form

f (x; n, p) = {\begin{cases} (\binom{n}{x}) p^{x} (1 - p)^{n - x} & x = 0, 1, 2, \dots, n \\ 0 & otherwise \end{cases}

We have that there’s a simplified form for $F$ being the cdf

F (k; n, p) = P (X \leq k) = I_{1 - p} (n - k, k + 1) = 1 - I_{p} (k + 1, n - k)

but in simple cases we can just sum over the integers $i \leq k$ . Where $I_{x} (a, b)$ represents the regularized incomplete beta function.

We have that

$E [X] = n p$
$Var [X] = n p (1 - p)$

Prop: Let $X_{1}, \dots, X_{n}$ be independent random variables each one with a distribution $Ber (p)$ . Then

\sum_{i = 1}^{n} X_{i} \sim bin (n, p)

And any random variable with distribution $bin (n, p)$ can be expressed as the sum above.
This means that any binomial random variable can be thought of as a sum of Bernulli Random Variables

With this we can get the probability generating function, getting that $$ G(t) = (1-p+pt)^n $$

the moment generating function

M (t) = (1 - p + p e^{t})^{n}

The characteristic function $$\phi(t) =(1-p+pe^{it})^n$$
Prop: Let $X$ and $Y$ be independent random variables, such that $X \sim bin (n, p)$ and $Y \sim bin (m, p)$ . We get that $X + Y \sim bin (n + m, p)$

Prop: Let $X$ be a random variable distributed by $bin (n, p)$ , then $n - X \sim bin (n, 1 - p)$

Prop: Let $X \sim bin (n, p)$ , then we have the following recurrent relation: $$P(X = i+1) = \frac{n-i}{i+1} \frac{p}{1-p} P(X = i)$$for $0 \leq i \leq n$ .