Sample Question #223 (statistics – Bayesian inference)

Suppose that *x *is normally distributed with mean *μ *and variance 4. Assume that the prior distribution of *μ *is also normal with mean 0 and variance 25. What is the posterior distribution of *μ *given the data point *x*?

[Taken from exercise 12.1 in Tsay’s *Analysis of Financial Time Series, *2nd ed.]

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ANSWER

If you say on your resume that you know statistics, you should have at least some passing knowledge of Bayesian analysis.

The posterior mean of μ is [4*0+1*25*x]/(4+1*25), where 1 is the number of observations we have, which is just one data point. The sample mean of one data point is just the data point itself, x. The posterior variance of μ is 4*25/(4+1*25).