ISI_2011 Question 29

[kangkan]

Hi..Can someone help me calculate the variance.

 A discrete random variable X takes value 0,1,2 …..n with frequency  (_0^n)C,(_1^n)C,…..(_n^n)C   .Find the variance .
I have calculated the mean as such
P(X)= (_x^n)C/2^n
Therefore E(X)=1C1/2^n+ 2C2/2^n…….nCn/2^n
                          = 1C1+2C2+…….nCn/2^n
                         =  n*2^(n-1)/2^n
                         = n/2
But E(X^2)=1C1/2^n +4C2/2^n +⋯…
I am not able to get an algebraic expression for this series.
I know the answer is n/4.

[economist]

You can answer if you recognize that X is binomial variate with parameters (n,1/2).
now variance of binomial is npq.
so variance is n/4

[kangkan]

Thanks