Question

If a variable X takes values 0, 1, 2,..., n with frequencies ⁿC₀, ⁿC₁, ⁿC₂ , ... , ⁿC_n, then write variance X.

Answer 1

$$\begin{gathered} \overline{x}=\frac{{\displaystyle \sum _{i=0}^n}{x}_i{f}_i}{{\displaystyle \sum _{i=0}^n}{f}_i}=\frac{0\times {}^{n}C_o+1\times {}^{n}C_1+...+n\times {}^{n}C_n}{{}^{n}C_o+{}^{n}C_1+...+{}^{n}C_n} \\ \Rightarrow \overline{x}=\frac{n\times 2^{n-1}}{{\displaystyle \frac{2^n}{n+1}}} \\ =\frac{n\left(n+1\right)}{2} \\ \therefore Var\left(X\right)={\sigma }^2 \\ =\frac{1}{n}\sum _{i=0}^n{\left(x_i-\overline{x}\right)}^2 \\ =\frac{1}{n}{\left[\left(0+1+2+....+n\right)-n\overline{x}\right]}^2 \\ \Rightarrow {\sigma }^2=\frac{1}{n}{\left[\frac{n\left(n+1\right)}{2}-\frac{n\times n\left(n+1\right)}{2}\right]}^2 \\ =\frac{1}{n}{\left[\frac{n\left(n+1\right)}{2}\left(1-n\right)\right]}^2 \\ =\frac{n^2}{4n}{\left(n+1\right)}^2{\left(n-1\right)}^2 \end{gathered}$$