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Suppose for this problem that "true IQ" is a meaningful concept rather than a reified social construct. Suppose that in the U.S. population, the distribution of true IQs is Normal with mean 100 and SD 15. A person is chosen at random from this population to take an IQ test. The test is a noisy measure of true ability: it's correct on average but has a Normal measurement error with SD 5. Let mean be the person's true IQ, viewed as a random variable, and let Y be his or her score on the IQ test. Then we have:
Y | mean ~N (mean , 5^2 ) mean ~ N(100, 15^2).

(a) Find the unconditional mean and variance of Y .

(b) Find the marginal distribution of Y . One way is via the MGF.

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Category:- Statistics and Probability
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