Binomial Distribution

$X\sim \text{Binomial}(n,p)$

A binomial random variable represents the number of successes out of $n$ independent trials that each have a probability $p$ of success

• $p_X(x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{n-x},x\in \{0,1,\dots ,n\}$
• $F_X(x)=$

• $M_X(t)=(1-p+p{e}^t{)}^n$
  This occurs because:
  1. $M_X(t)=\sum _{x=0}^n{e}^{tx}{p}_X(x)$
  2. $M_X(t)=\sum _{x=0}^n{e}^{tx}{p}^x(1-p{)}^{n-x}$
  3. $M_X(t)=\sum _{x=0}^n(e^{t}p{)}^x(1-p{)}^{n-x}$
  4. By Binomial theorem: $M_X(t)=(1-p+p{e}^t{)}^n$

• $\mathbb{E}[X]=np$
• $\text{var}(X)=np(1-p)$
• $⟨x^2⟩={\sigma }^2+{\mu }^2$
• A binomial distribution describes the number of successes when running $n$ independent trials where the probability of a success for each trial is $p$

Applications

• Spin Magnet System