Binomial Distribution
#Math #Physics
$X\sim\text{Binomial}(n,p)$
A binomial random variable represents the number of successes out of $n$ independent trials that each have a probability $p$ of success
$p_X(x)={n\choose x}p^x(1-p)^{n-x},~x\in{0,1,\ldots,n}$
$F_X(x) =$
$M_X(t)=(1-p+pe^t)^n$
This occurs because:- $M_X(t)=\displaystyle\sum_{x=0}^ne^{tx}p_X(x)$
- $M_X(t)=\displaystyle\sum_{x=0}^ne^{tx}p^x(1-p)^{n-x}$
- $M_X(t)=\displaystyle\sum_{x=0}^n(e^tp)^x(1-p)^{n-x}$
- By Binomial theorem: $M_X(t)=(1-p+pe^t)^n$
$\mathbb{E}[X]=np$
$\text{var}(X) = np(1 - p)$
$\displaystyle {\left\langle{x^{2}}\right\rangle}=\sigma ^{2}+\mu ^{2}$
A binomial distribution describes the number of successes when running $n$ independent trials where the probability of a success for each trial is $p$