The mean of the values 0, 1, 2, dots, n havin | vedclass

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(D) The weighted mean is given by $\bar{x} = \frac{\sum_{r=0}^{n} r \cdot {^nC_r}}{\sum_{r=0}^{n} {^nC_r}}$.
The numerator is $\sum_{r=0}^{n} r \cdot

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$\frac{n}{2}$

Vedclass · Answer

(D) The weighted mean is given by $\bar{x} = \frac{\sum_{r=0}^{n} r \cdot {^nC_r}}{\sum_{r=0}^{n} {^nC_r}}$.
The numerator is $\sum_{r=0}^{n} r \cdot {^nC_r} = \sum_{r=1}^{n} r \cdot \frac{n}{r} \cdot {^{n-1}C_{r-1}} = n \sum_{r=1}^{n} {^{n-1}C_{r-1}} = n \cdot 2^{n-1}$.
The denominator is $\sum_{r=0}^{n} {^nC_r} = 2^n$.
Therefore, the mean is $\bar{x} = \frac{n \cdot 2^{n-1}}{2^n} = \frac{n}{2}$.

The mean of the values $0, 1, 2, \dots, n$ having corresponding weights $^nC_0, ^nC_1, ^nC_2, \dots, ^nC_n$ respectively is

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