Question

If a variable takes values $0,1,2,3,...,n$ with frequencies
$1,C(n,1),C(n,2),C(n,3),...,C(n,n)$ respectively, then the arithmetic mean is

• A. $2n$
• B. $n+1$
• C. $n$
• D. $\frac{n}{2}$

Answer 1

Answer: (D)

$\frac{n}{2}$

mean $=\frac{0\times 1+1\times C(n,1)+2\times C(n,2)+......+n\times C(n,n)}{1+C(n,1)+C(n,2)+......+C(n,n)}$

$=\frac{1\times C(n,1)+2\times C(n,2)+......+n\times C(n,n)}{C(n,0)+C(n,1)+C(n,2)+......+C(n,n)}$

$=\frac{{\sum }_{r=1}^{n}r\times C(n,r)}{2^n}$

$=\frac{{\sum }_{r=1}^{n}r\times \frac{n}{r}\times C(n-1,r-1)}{2^n}$

$=\frac{n\times 2^{n-1}}{2^n}=\frac{n}{2}$