Tuesday, November 19, 2013

The variance of two estimators

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 x1, x2 …. xn are binary with two outcomes 1 with probability p and 0 with probability 1-p.  

 What is the Var(nx1)? 

What is the Var (x1+x2+…xn)?  

What is the meaning of the difference in the variances for the two estimators?


 The var(nx1) is n2*var(x1), which is n2*p*(1-p)

The Var(x1+x2+…xn) is var(x1) +var(x2) +… var(xn), which is n*p*(1-p).

The derivation of the variance of the binary variable  — Va(xi) =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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