Variance of the number of ice-cream scoops ordered

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At an ice-cream stand, the number of scoops a child orders is a random number X: Scoops 0 1 2 ------------ Chance 25% 50% 25% What is the variance of the number of scoops?

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Variance measures how far the scoop counts spread out around their average. Use the shortcut \mathrm{Var}(X) = E[X^2] - \big(E[X]\big)^2, which reads: average of the squares minus the square of the average. Mean. Because the pattern is symmetric around 1, the average is exactly 1: E[X] = 0(0.25) + 1(0.50) + 2(0.25) = 0 + 0.5 + 0.5 = 1. Average of the squares. E[X^2] = 0^2(0.25) + 1^2(0.50) + 2^2(0.25) = 0 + 0.5 + 1.0 = 1.5. Subtract. \mathrm{Var}(X) = 1.5 - 1^2 = 0.5. The standard deviation is \sqrt{0.5} \approx 0.707 scoops. Takeaway. When a distribution is symmetric around a whole number, t…

Variance of the number of ice-cream scoops ordered

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