We have described the beta distribution for a population proportion as an example of a conjugate prior when the likelihood function, p(x | ),

We have described the beta distribution for a population proportion µ as an example of a conjugate prior when the likelihood function, p(x | π), is binomial. Under certain conditions, the normal distribution also serves as a conjugate prior for the population mean µ. It can be shown that if
• p(µ), the prior distribution for m, is normal with mean u and variance τ2 [i. e., µ is N(θ, τ2)] and
• p(xi | µ), the likelihood function for a single observation xi, is N(µ, π2), for i = 1, 2, … , n, then the posterior distribution for m, p(µ | x1, x2, … , xn) or simply p(µ | data), is also normal, with posterior mean µ* and variance σ2* given by
µ* = σ2θ + nτ2 / σ2 + nτ2 and σ2* = σ2τ2 / σ2 + nτ2
Where x is the sample mean of the observations x1, x2, … , xn. Suppose the lengths (in inches) of a particular species of salmon are normally distributed with unknown mean µ but known variance 36 inches2. The prior distribution of µ is assumed to be normal with mean θ = 32 inches and variance τ2 = 25 inches2. Ten salmon are caught, and their lengths are 19, 26, 18, 26, 31, 31, 32, 36, 33, and 21 inches.
a. Find the posterior distribution of the mean salmon length, p(µ | data), and provide the values of the posterior mean, µ*, and posterior variance, σ2*.
b. Use statistical software or a standard normal table to find the posterior probability that the average fish length is greater than 35 inches.