Exercise 1550Combinatorics › Binomial coefficientsL1Added 3 Nov 2026

Moments of the binomial row

#moments#binomial row#mean

Statement

Treat the $n$ -th row of Pascal's triangle as a probability distribution after normalizing by $2^n$ . Compute its mean.

Hint

Mean = $\sum_k k\binom{n}{k}/2^n=n/2$ .

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Mean. $\sum_k k\binom{n}{k}/2^n=\frac{n\cdot 2^{n-1}}{2^n}=\frac{n}{2}$ .

Variance. $\sum_k k^2\binom{n}{k}/2^n=\frac{n(n+1)2^{n-2}}{2^n}=\frac{n(n+1)}{4}$ , so variance = $\frac{n(n+1)}{4}-\frac{n^2}{4}=\frac{n}{4}$ .

Conclusion. This is $\mathrm{Bin}(n,1/2)$ .