I originally created this problem 30 years ago when I taught statistics at Kansas State University. It helps explain the concept of variance.

**Question**: Random variables x

_{1,}x

_{2}…. x

_{n}are binary with two outcomes 1 with probability p and 0 with probability 1-p.

What is the Var(nx

_{1})?
What is the Var (x

_{1}+x_{2}+…x_{n})?
What is the meaning of the difference in the variances for the two estimators?

**Answer**:

The var(nx

_{1}) is n^{2}*var(x_{1}), which is n^{2}*p*(1-p)
The Var(x

_{1}+x_{2}+…x_{n}) is var(x_{1}) +var(x_{2}) +… var(x_{n}), which is n*p*(1-p).
The derivation of the variance of the binary variable — Va(x

_{i}) =p*(1-p) — can be found at
The point that I was trying to make with this problem is that the variance of the sum of n identical random variables is smaller than the variance of n multiplied by the outcome from one random variable. This is because there is more information in N observations than one observation assuming of course that N is greater than one and all observations are from the same distribution.

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