Binomial Distribution

Definition

$X\sim B(n,p)$

Models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success

$x$ is the amount of success outcomes

• pmf: $\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{n-x}$, $x\in 0,1,2,\dots ,n$
• Mean: $np$
• Variance: $np(1-p)$
• mgf: $(1-p+p{e}^t{)}^n$