Combined revenue in a range

Question

Two stores have independent daily revenues. Store 1's revenue is normally distributed with mean 200 and standard deviation 9 (in thousands). Store 2's revenue has mean 300 and standard deviation 12.

What is the probability that their combined revenue for the day falls between 485 and 530?

Accepted Answer

The sum of the two independent normal revenues is normal. Add the means, and add the variances (squared standard deviations). Mean of the combined revenue: 200 + 300 = 500. Variance of the combined revenue: 9^2 + 12^2 = 81 + 144 = 225, so the standard deviation is \sqrt{225} = 15. The total is normal with mean 500 and standard deviation 15. For a range, standardize both endpoints: Lower, 485: \;z = \dfrac{485 - 500}{15} = -1. Upper, 530: \;z = \dfrac{530 - 500}{15} = 2. The probability of landing between them is the area below the upper z minus the area below the lower z: P(485 \le 	ext{tota…

Combined revenue in a range

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