Variance of the number of train tickets a commuter buys

Question

At a station kiosk, the number of tickets a commuter buys in one visit is a random number $X$ :

Tickets	1	2	3	4
Chance	40%	30%	20%	10%

What is the variance of the number of tickets bought?

Accepted Answer

Variance tells us how much the order size swings around its average. Use \mathrm{Var}(X) = E[X^2] - \big(E[X]\big)^2. Mean. Weight each count by its chance: E[X] = 1(0.4) + 2(0.3) + 3(0.2) + 4(0.1) = 0.4 + 0.6 + 0.6 + 0.4 = 2.0. Average of the squares. Square each value first: E[X^2] = 1(0.4) + 4(0.3) + 9(0.2) + 16(0.1) = 0.4 + 1.2 + 1.8 + 1.6 = 5.0. Subtract. \mathrm{Var}(X) = 5.0 - 2.0^2 = 5.0 - 4.0 = 1.0. The standard deviation is \sqrt{1} = 1 ticket, so orders sit around 2 tickets, give or take about 1. Takeaway. Even with values starting at 1 instead of 0, the recipe is unchanged: mean, …

Variance of the number of train tickets a commuter buys

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