Two components stacked in height

Question

Two parts are stacked. The bottom part's height is normally distributed with mean 100 mm and standard deviation 5 mm. The top part's height is normally distributed with mean 150 mm and standard deviation 12 mm. The two are independent.

What is the probability that the stacked height exceeds 263 mm?

Accepted Answer

The sum of two independent normal heights is normal. Add the means directly, and add the variances (the squares of the standard deviations). Mean of the stack: 100 + 150 = 250 	ext{ mm}. Variance of the stack: 5^2 + 12^2 = 25 + 144 = 169, so the standard deviation is \sqrt{169} = 13 mm. The stack height is normal with mean 250 mm and standard deviation 13 mm. Convert the 263 mm cutoff to a z-score: z = \frac{263 - 250}{13} = 1. The area beyond z = 1 is about 1 - 0.8413 = 0.1587, so P(	ext{stack} 263) \approx 0.1587. The 5, 12, 13 combination is a classic reminder that spreads add like the l…

Two components stacked in height

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