If a variable $X$ takes values $0, 1, 2, ... , n$ with frequencies ${ }^n{C}_0{,}^n{C}_1{,}^n{C}_2,\dots {.}^n{C}_n$ respectively, then S.D $=$

detailed solution

Correct option is C

We have,    σ2=1N∑i=1n fixi2−1n∑i=1n fixi2∴ σ2=12n∑r=0n Crr2−12n∑r=0n Crr2∵N=∑r=0n Crn=2nNow,     ∑r=0n rnCr=∑r=0n rnrn−1Cr−1⇒ ∑r=0n rnCr=n∑r=1n−1 Cr−1=n2n−1and,      ∑r=0n r2nCr=∑r=0n {r(r−1)+r}nCr⇒ ∑r=0n r2nCr=∑r=0n r(r−1)nCr+∑r=0n rnCr⇒ ∑r=0n r2nCr=∑r=2n r(r−1)n−1r−1n−2r−1Cr−2+∑r=1n rn−nn−1Cr−1r⇒ ∑r=0n r2nCr=n(n−1)∑r=2n n−2Cr−2+n∑r=1n n−1Cr−1⇒ ∑r=0n r2nCr=n(n−1)2n−2+n2n−1∴ σ2=n(n−1)2n−2+n×2n−12n−n×2n−12n2⇒ σ2=n(n−1)4+n2−n24=n4⇒σ=n2