Standard deviation of daily profit from units sold

Question

A stall's daily profit in dollars is P = 8\,X - 100, where X is the number of units sold that day, and 100 is the fixed daily cost. The number of units sold has variance 25 (so its standard deviation is 5). What is the standard deviation of the daily profit?

Accepted Answer

Break the formula into its two effects. The -100 is a fixed cost subtracted every single day; taking the same amount off every outcome slides them all down together and does not change how spread out they are. The 	imes 8 is a markup that stretches each unit sold into 8 dollars, and that stretching does widen the spread. The rule for P = aX + b is \mathrm{Var}(P) = a^2\,\mathrm{Var}(X), \qquad \mathrm{SD}(P) = a\,\mathrm{SD}(X). Apply it. Here a = 8 and b = -100, and the fixed cost drops out: \mathrm{SD}(P) = 8 	imes 5 = 40 	ext{ dollars}. As a variance check, \mathrm{Var}(P) = 8^2 	imes …

Standard deviation of daily profit from units sold

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