AP Statistics Curriculum 2007 Bayesian Normal

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Normal Example:

It is known that the speedometer that comes with a certain new sports car is not very accurate, which results in an estimate of the top speed of the car of 185 mph, with a standard deviation of 10 mph. Knowing that his car is capable of much higher speeds, the owner took the car to the shop. After a checkup, the speedometer was replaced with a new one, which gave a new estimate of 220 mph with a standard deviation of 4 mph. The errors are assumed to be normally distributed. We can say that the owner S’s prior beliefs about the top speed of his car were represented by:

µ ~ N(μ0, φ0) = µ ~ N(185,102)

We could then say that the measurements using the new speedometer result in a measurement of:

x ~ N(μ, φ) = x ~ N(µ,42)

We note that the observation x turned out to be 210, and we see that S’s posterior beliefs about µ should be represented by:

µ | x ~ N(μ1, φ1)

where (rounded)

φ1 = (10 − 2 + 4 − 2) − 1 = 14 = 42
μ1 = 14(185 / 102 + 220 / 42) = 218

Therefore, the posterior for the top speed is:

μ | x ~ N(218,42)

Meaning 218 +/- 4 mph.

If the new speedometer measurements were considered by another person S’ who had no knowledge of the readings from the first speedometer, but still had a vague idea (from knowledge of the stock speedometer) that the top speed was about 200 +/- 30 mph, Then:

μ ~ N(200,302)

Then S’ would have a posterior variance:

φ1 = (30 − 2 + 4 − 2) − 1 = 16 = 42

S’ would have a posterior mean of:

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