Celsius to Fahrenheit for a normal reading

Question

A sensor's temperature reading is X \sim N(	ext{mean } 20,\ 	ext{variance } 4) in degrees Celsius. You convert to Fahrenheit with F = 1.8X + 32. What are the mean and variance of F?

Accepted Answer

A linear rescaling aX + b shifts and stretches a normal, and it stays normal. The rules: - Mean: \mathbb{E}[aX+b] = a\,\mathbb{E}[X] + b. The shift b moves the center; the scale a multiplies it. - Variance: \operatorname{Var}(aX+b) = a^2\operatorname{Var}(X). Adding a constant b does not change spread at all; multiplying by a scales the variance by a^2 (and the standard deviation by a). Here a = 1.8, b = 32: \mathbb{E}[F] = 1.8(20) + 32 = 68, \qquad \operatorname{Var}(F) = 1.8^2 	imes 4 = 3.24 	imes 4 = 12.96. Standard deviation 1.8 	imes 2 = 3.6 degrees F. The +32 had no effect on spread.…

Celsius to Fahrenheit for a normal reading

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