Binomial Distribution

We use the Binomial Distribution to answer the following question:

• Given $n$ independent trials and a probability of success $p$ per trial, what is the probability that we have $k$ total successes, i.e. $P(X=k)$?

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 $p$ $X\sim Bin(n,p)$

Binomial Distribution - Bernoulli Trials (Definition)

If $X_1,X_2,\dots ,X_n$ are independent $Ber(p)$ and $X=X_1+X_2+\cdots +X_n$ $X\sim Bin(n,p)$

Suppose $X\sim Bin(n,p)$

1. Support of $X:\{0,1,2,\dots ,n\}$
2. PMF: $P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$ where

• $k$ is the number of successes
• $n$ is the number of trials
• $p$ is the probability of success for 1 trial

Expectation and Variance

• $E(X)=np$
• $Var(X)=np(1-p)$