Derive the posterior probability-density function P( ω 1 I x) for the likelihood functions...

Derive the posterior probability-density function P(ω_{1} I x) for the likelihood functions defined in problem 1.44 but having different variance

Problem 1.44

Find the posterior probability-density functions “(ω_{1} I x) and P(ω_{2} I x) for the two equiprobable one-dimensional normally distributed classes given by likelihood functions (class-conditional probability-density functions that are also known as data generation mechanisms)

(Hint: Start with Bayes’ rule, plug in the given likelihoods, and find the desired posterior probability-density functions in terms of distribution means and standard deviation.)