Question

The mean of the observations $0,1,2,3,\dots ,n$ having corresponding weight's ${}^n{C}_0,{}^n{C}_1,{}^n{C}_2,\dots ,{}^n{C}_n$ respectively is:

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

Answer 1

The correct option is D $\frac{n}{2}$

$\text{Weighted mean}=\frac{\sum w_i{x}_i}{\sum w_i}$
$=\frac{\sum _{r=0}^{n}r\cdot {}^n{C}_r}{\sum _{r=0}^n{}^n{C}_r}$
$=\frac{n\cdot 2^{n-1}}{2^n}$
$=\frac{n}{2}$