1. Tardigrade
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  3. Mathematics
  4. The mean of the values 0,1,2, ldots, n having corresponding weight nC0, nC1, nC2, ldots, nCn respectively, is

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Q. The mean of the values $0,1,2,\ldots, n$ having corresponding weight $^{n}C_{0}, ^{n}C_{1}, ^{n}C_{2}, \ldots, ^{n}C_{n}$ respectively, is

UPSEEUPSEE 2012

Solution:

The required mean is
$\bar{x}=\frac{0 \cdot 1+1 \cdot{ }^{n} C_{1}+2 \cdot{ }^{n} C_{2}+3 \cdot{ }^{n} C_{3}+\ldots+n \cdot{ }^{n} C_{n}}{1+{ }^{n} C_{1}+{ }^{n} C_{2}+\ldots+{ }^{n} C_{n}}$
$=\frac{\displaystyle\sum_{r=0}^{n} r \cdot{ }^{n} C_{r}}{\displaystyle\sum_{r=0}^{n}{ }^{n} C_{r}}=\frac{\displaystyle\sum_{r=1}^{n} r \cdot \frac{n}{r} \cdot{ }^{n-1} C_{r-1}}{\displaystyle\sum_{r=0}^{n}{ }^{n} C_{r}}$
$=\frac{n \displaystyle\sum_{r=1}^{n}{ }^{n-1} C_{r-1}}{\displaystyle\sum_{r-0}^{n}{ }^{n} C_{r}}$
$=\frac{n \cdot 2^{n-1}}{2^{n}}=\frac{n}{2}$