Mean and Variance of Binomial Distribution ( Formulae without proof )

Mean and Variance of Binomial Distribution ( Formulae without proof )

These results are stated here without proof. If $X\sim B(n,p)$, then the mean (expected value) of $X$ is denoted $\mu$ or $E(X)$ and given by

The variance is denoted $\text{Var}(X)$ and given by

The standard deviation is denoted $SD(X)$ or $\sigma_X$, and is the square root of the variance:

In-text example. If $X\sim B(10, 0.4)$, here $n=10$, $p=0.4$, so $q=1-0.4=0.6$. Then $E(X)=np=10\times0.4=4$ and $\text{Var}(X)=npq=10\times0.4\times0.6=2.4$.

Worked examples.

Ex.1 - The p.m.f. $P(X=x)={}^{4}C_x\left(\tfrac59\right)^x\left(\tfrac49\right)^{4-x}$, $x=0,1,2,3,4$, is a binomial distribution with $n=4$, $p=\tfrac59$, $q=\tfrac49$ (matching the standard ${}^{n}C_xp^xq^{n-x}$ pattern). So $E(X)=np=4\times\tfrac59=\tfrac{20}{9}$ and $\text{Var}(X)=npq=4\times\tfrac59\times\tfrac49=\tfrac{80}{81}$. …