If a variable x takes values 0,1,2,..,n with frequencies proportional to the binomial coefficients `.^(n)C_(0),.^(n)C_(1),.^(n)C_(2),..,.^(n)C_(n)`, then var (X) is

Options

A. `(n^(2)-1)/(12)`
B. `(n)/(2)`
C. `(n)/(4)`
D. None of these

Answer

C

Explanation

We have
`overline(X)=(0^(n)C_(0)+1^(n)C_(1)+2^(n)C_(2)+..+n^(n)C_(n))/(.^(n)C_(0)+.^(n)C_(1)+.^(n)C_(2)+..+.^(n)C_(n))=(underset(r=0)overset(n)(sum r^(n)C_(r )))/(underset(r=0)overset(n)(sum .^(n)C_(r )))`
`=(1)/(2^(n))underset(r=1)overset(n)(sum r)(n)/(r ).^(n-1)C_(r-1) " " [because underset(r=0)overset(n)(sum).^(n)C_(r )=2^(n), .^(n)C_(r )=(n)/(r ).^(n-1)C_(r-1)]`
`=(n)/(2^(n))underset(r=1)overset(n)(sum).^(n-1)C_(r-1)=(n)/(2^(n))2^(n-1)=(n)/(2)[because underset(r=1)overset(n)(sum).^(n-1)C_(r-1)=2^(n-1)]`
and `(1)/(N)sum f_(i)x_(1)^(2)=(1)/(2^(n))sum r^(2) .^(n)C_(r )=(1)/(2^(n)) underset(r=0)overset(n)(sum)[r(r-1)+r].^(n)C_(r )`
`=(1)/(2^(n)){underset(r=0)overset(n)(sum)r-(r-1).^(n)C_(r )+underset(r=0)overset(n)(sum r) .^(n)C_(r )}`
`=(1)/(2^(n)){underset(r=2)overset(n)(sum)r(r-1)(n)/(r )(n-1)/(r-1) .^(n-2)C_(r-2)+underset(r=1)overset(n)(sum r) (n)/(r ).^(n-1)C_(r-1)}`
`=(1)/(2^(n)){n(n-1)2^(n-2)+n2^(n-1)}=(n(n-1))/(4)+(n)/(2)`
`therefore "Var"(X)=(1)/(N)sum f_(i)x_(i)^(2)-overline(X^(2))=(n(n-1))/(4)+(n)/(2)-(n^(2))/(4)=(n)/(4)`