3. Let #1,..., #, ber populations (r > 2) with density functions p (), ..., p,(x) and a priori probabilities 91, ...,qr, respectively. We classify an observation a as coming from nj if ac ER;, where R1, ..., Rr are chosen mutually exclusive and exhaustive regions in Rm. Note that by Bayes formula the conditional probability of the observation coming from n; is If we classify the observation as from my, the expected loss is 4-1 EX-1 9kPK (I) (i) Let Li(x) = Eqp:(a). Show that E Li(x) da j=1 is minimized for the following choice of the classification regions: R = (x : q;p;(x) > qp.(x), i=1,...,r, if j). (1) for j = 1, ...,". (If the equality happens in the above definition of the classification regions, the observation can be classified in any one of the corresponding populations.)