Problem 5 Consider the probability density function of the one-dimensional normal distribution with mean ?and variance ? exp (- (x-?)2 ) . Let us also consider n real-valued observations; they are denoted by - (T1,32,...,xn). We further suppose the following e Each observation zi (where i = 1,2, m) is independently distributed, following the one- dimensional normal distribution with mean ? and variance ? . Variance ? is known. The prior distribution of ? is the one-dimensional normal distribution with m ean mo and variance ? Answer the following questions. (1) Given an observation let p(u | 1,o0, mo, Po) denote the probability density function of the Bayes posterior distribution of the random variable . This distribution is a one-dimensional normal distribution, and its probability density function is given by N(#1 mi, 1). Express mean mi and variance ?, using z1,0 , mo and P You can use the following equality: (2) Let n 2. Given observations x1m-(X1, 22, An), let p(? | ?1:n, 05, mo, p') denote the probability density function of the Bayes posterior distribution of the random variable u. This distribution is a one-dimensional normal distribution, and its the probability density function is given by N(? mn, p), Express mean mn and variance , using n,al:n, ??, mo and (3) For the Bayes posterior distribution p(? | 21:n,?,mo,PG) of Question (2), show that the following equality holds mn-1, Pn-1 Problem 5 Consider the probability density function of the one-dimensional normal distribution with mean ?and variance ? exp (- (x-?)2 ) . Let us also consider n real-valued observations; they are denoted by - (T1,32,...,xn). We further suppose the following e Each observation zi (where i = 1,2, m) is independently distributed, following the one- dimensional normal distribution with mean ? and variance ? . Variance ? is known. The prior distribution of ? is the one-dimensional normal distribution with m ean mo and variance ? Answer the following questions. (1) Given an observation let p(u | 1,o0, mo, Po) denote the probability density function of the Bayes posterior distribution of the random variable . This distribution is a one-dimensional normal distribution, and its probability density function is given by N(#1 mi, 1). Express mean mi and variance ?, using z1,0 , mo and P You can use the following equality: (2) Let n 2. Given observations x1m-(X1, 22, An), let p(? | ?1:n, 05, mo, p') denote the probability density function of the Bayes posterior distribution of the random variable u. This distribution is a one-dimensional normal distribution, and its the probability density function is given by N(? mn, p), Express mean mn and variance , using n,al:n, ??, mo and (3) For the Bayes posterior distribution p(? | 21:n,?,mo,PG) of Question (2), show that the following equality holds mn-1, Pn-1