Binomial Distribution

We use the Binomial Distribution to answer the following question:

  • Given independent trials and a probability of success per trial, what is the probability that we have total successes, i.e. ?
title: Binomial Distribution (Definition)
Suppose we have $n$ independent trials, each with two possible  
outcomes:  
- success: probability $p$
- failure: probability $1 − p$
 
If $X =$ the number of successes in $n$ trials; then $X$ follows a  
Binomial Distribution with parameters $n$ and $p$
$$X ∼ Bin(n, p)$$
 
title: Binomial Distribution - Bernoulli Trials (Definition)
If $X_{1}, X_{2}, \dots, X_{n}$ are independent $Ber(p)$ and $X =X_{1}+X_{2}+\dots+X_{n}$
$$X ∼ Bin(n, p)$$

Suppose

  1. Support of
  2. PMF: where
  • is the number of successes
  • is the number of trials
  • is the probability of success for 1 trial

Expectation and Variance