Question

A variable takes the values $0,1,2,3,....n$ with frequencies ${}^n{c}_0,{{}^{n}c}_1,{{}^{n}c}_2,{{}^{n}c}_3,......{{}^{n}c}_n$ the mean $\overline{x}=$

Answer 1

We have, $(1+x{)}^n{=}^n{C}_0{+}^n{C}_1.x+.....{+}^n{C}_n{x}^n$

Now differentiating both sides with respect to $x$ we get,

$n(1+x{)}^{n-1}{=}^n{C}_1+2{}^n{C}_2.x+.....+n{}^n{C}_n.x^{n-1}$

Now putting $x=1$ both sides we get,

$n{2}^{n-1}{=}^n{C}_1+2{}^n{C}_2+...+n{}^n{C}_n$......(1).

Now for the problem,

$\overline{x}=\frac{0+1.{}^n{C}_1+2.{}^n{C}_2+.....+n.{}^n{C}_n}{0+1+2+.....+n}$

or, $\overline{x}=\frac{n{2}^{n-1}}{\frac{n(n+1)}{2}}$

or, $\overline{x}=\frac{2^n}{n+1}$.