Variance of a customer's extra espresso-shot order

Question

At a coffee bar, the number of extra espresso shots a customer adds to a drink is a random number $X$ with this distribution:

Extra shots	0	1	2	3
Chance	20%	40%	30%	10%

What is the variance of the number of extra shots, and its standard deviation?

Accepted Answer

Variance is just a number that says how spread out the shot counts tend to be around their average. The plan is two steps: first find the average (the mean), then find the average of the squared distances from that mean. The handy shortcut is \mathrm{Var}(X) = E[X^2] - \big(E[X]\big)^2, which in words says: take the average of the squares, then subtract the square of the average. Step 1: the mean. Multiply each value by its chance and add: E[X] = 0(0.2) + 1(0.4) + 2(0.3) + 3(0.1) = 0 + 0.4 + 0.6 + 0.3 = 1.3. So a typical order is about 1.3 extra shots. Step 2: the average of the squares. Squa…

Variance of a customer's extra espresso-shot order

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