If a variable x takes values 0,1,2,…,n with frequencies  proportional to the binomial coefficients nC0,nC1,nC2,… nCn , then var⁡(x) is

We have x¯=0nC0+1nC1+2nC2+⋯+nnCn nC0+nC1+nC2+⋯+nCn=∑r=0n rnCr∑r=0n nCr    =12n∑r=1n rnrn−1Cr−1 ∵∑r=0n nCr=2n;nCr=nrn−1Cr−1     =n2n∑r=1n n−1Cr−1=n2n2n−1=n2 ∵∑r=1n n−1Cr−1=2n−1 and 1N∑fixi2=12n∑r=0nr2nCr=12n∑r=0n [r(r−1)+r]nCr=12n∑r=0n r(r−1) Cr+∑r=0n r Cr=12n∑r=2n r(r−1)nrn−1r−1n−2Cr−2+∑r=1n rnrn−1Cr−1=12nn(n−1)2n−2+n2n−1=n(n−1)4+n2∴Var⁡(X)=1N∑fixi2−x¯2=n(n−1)4+n2−n24=n4