Suppose a variable assumes the values 0,1,2,3,...n with frequencies proportional to binomial coefficients `nc0,nc1,nc2,nc3,...,,ncn.` Find the mean of variables.

Explanation

Required mean will be,

`M = (0*C(n,0)+1*C(n,1)+2*C(n,2)+...n*C(n,n))/(n+1)`

`=>M = (1*C(n,1)+2*C(n,2)+...n*C(n,n))/(n+1)->(1)`

Now, we know,

`(1+x)^n = C(n,0) +C(n,1)x+C(n,2)x^2+...C(n,n)x^n`

Differentating both sides w.r.t. `x`,

`n(1+x)^(n-1) = 1*C(n,1)+2*C(n,2)x+...n*C(n,n)x^(n-1)`

For, `x = 1`,

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