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−1and 1N∑fixi2=12n∑r2nCr=12n∑r=0n [r(r−1)+r]nCr     =12n∑r=0n r−(r−1)nCr+∑r=0n rnCr     =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