2.39 ( ) Starting from the results (2.141) and (2.142) for the posterior distribution of the mean of a Gaussian random variable, dissect out the contributions from the first N − 1 data points and hence obtain expressions for the sequential update of

μN and σ2N

. Now derive the same results starting from the posterior distribution p(μ|x1, . . . , xN−1) = N(μ|μN−1, σ2N

−1) and multiplying by the likelihood function p(xN|μ) = N(xN|μ, σ2) and then completing the square and normalizing to obtain the posterior distribution after N observations.