I would like to understand this in steps and how can I identify posterior mean and variance

Suppose that we observe x ? N(?, 1), where ? is the mean of the normal distribution and 1 is the variance of the normal distribution.

8. Suppose that we observe x ~ N(0, 1), where 0 is the mean of the normal distribution and variance of the normal distribution. In practice, we often use a conjugate prior, say, given TT (0) x - 1 V27 with / known. With such a prior distribution, find the posterior distribution, the posterior mean and the p variance. Solution T. (0|z) x exp(-(0 -z)2/2- (0 -1)2/2) 7(0|x) x exp(-(202 - 2(x + /)0 + 22 + 12)/2) = exp - 2(02 - (2 + 1) 0) + 22+ 12 2 2 0 - (2+1)) 2 a exp 2 This implies that 2 0 - (2+1) ) 2 T(0|z) = exp 2 is a normal distribution. It has posterior mean (x + /)/2 and posterior variance 1/2