If a variable X takes values 0,1,2, , n with frequencies ^ n C _0, ^n C _1, ^ n C _2, , ^ n C _ n , then write variance X .

x = _ i=0 ^ n x_if_i _ i=0 ^ n f_i = 0 ^nC_o+1 ^nC_1+...+n ^nc_n ^nC_o+^nC_1+.....+^nc_n x = n 2^ n-1 2^n n+1 = n(n+1) 2 (X)= ^2 =1 n ^ n _ i=0 (x_i- x )^2 =1 n [(0+1+2+....+n)-n x ]^2 ^2=1 n [ n(n+1) 2 - n (n+1) 2 ]^2 =1 n [ n(n+1) 2 (1-n) ]^2 = n^2 4n (n+1)^2(n-1)^2

If a variable X takes values 0,1,2, , n with frequencies ^ n C _0…

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Question

If a variable X takes values $0,1,2, \ldots, n$ with frequencies ${ }^{ n } C _0,{ }^n C _1,{ }^{ n } C _2, \ldots,{ }^{ n } C _{ n }$, then write variance X .

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Answer

$\overline{\text{x}}=\frac{\sum\limits_{\text{i}=0}^{\text{n}}\text{x}_\text{i}\text{f}_\text{i}}{\sum\limits_{\text{i}=0}^{\text{n}}\text{f}_\text{i}}=\frac{0\times^\text{n}\text{C}_\text{o}+1\times^\text{n}\text{C}_1+...+\text{n}\times^\text{n}\text{c}_\text{n}}{^\text{n}\text{C}_\text{o}+^\text{n}\text{C}_1+.....+^\text{n}\text{c}_\text{n}}$ $\Rightarrow\overline{\text{x}}=\frac{\text{n}\times2^{\text{n}-1}}{\frac{2^\text{n}}{\text{n}+1}}$ $=\frac{\text{n}(\text{n}+1)}{2}$ $\therefore\text{Var}(\text{X})=\sigma^2$ $=\frac{1}{\text{n}}\sum\limits^{\text{n}}_{\text{i}=0}\big(\text{x}_\text{i}-\overline{\text{x}}\big)^2$ $=\frac{1}{\text{n}}[(0+1+2+....+\text{n})-\text{n}\overline{\text{x}}]^2$ $\Rightarrow\sigma^2=\frac{1}{\text{n}}\Big[\frac{\text{n}(\text{n}+1)}{2}-\frac{\text{n}\times\text{n}(\text{n}+1)}{2}\Big]^2$ $=\frac{1}{\text{n}}\Big[\frac{\text{n}(\text{n}+1)}{2}(1-\text{n})\Big]^2$ $=\frac{\text{n}^2}{4\text{n}}(\text{n}+1)^2(\text{n}-1)^2$

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