Binomial distribution

Definition: $X\sim B(n,p)$

• When event $X$ has binary response, either fail or success
  • Discrete random variable
• Condition:
  • The same experiment is repeated a fixed number of times
  • There are only 2 possible outcomes, success and failure
  • The repeated trials are independent, so that the probability of success remains the same for each trial

Probability mass function

• $P(X=x)=\left(\begin{array}{c}n\\ x\end{array}\right)p^x{q}^{n-x}$

Expected value $E[X]=np$

• Proof: ^72433b
  • Let $X$ be a binomial random variable with parameters $n$ and $p$
  • Then $X=\sum _{i=1}^n{X}_i$ where $X_i\{\begin{array}{ll}1& \text{if the ith trial success}\\ 0& \text{otherwise}\end{array}$
  • $X_i\sim B(1,p)\to E[X_i]=1\ast p+0\ast (1-p)=p$
  • $E[X]=\sum E[X_i]=np$

Variance $Var(X)=npq$

• Proof:
  • Similarly for above, $Var(X_i)=p-p^2$
  • $Var(X)=Var(X_1)+...+Var(X_n)=n(p-p^2)=n.p.(1-p)$

Sum of independent Binomial distribution:

• Sum of independent variables
• $X\sim B(n,p)$ and $Y\sim B(m,p)$ then $X+Y\sim B(n+m,p)$