1. Let Ho and Hi be two events. Let pi(X1, ...,Xn) be the conditional density of A1, ..., An given H; occurred. (a) Assume that given Hi, {X}}})_ is a sequence of i.i.d. Gaussian random variables with mean (-1)'VE and variance No/2. Use the Chernoff bound to show that P{pi(X) > po(X) |Ho} SenE/No (b) Now let X1, ...,An be independent discrete random variables taking values +1 and -1 with P{Xi =+1[Ho} = P, P{X; =-1\\Ho} = 1-p, P(X = -1H1} = P, P{X =+1 H1} = 1-p. Thus if the number of components of x equal to +1 is d then po(x) = p(1 -p)"-d and Pi(x) = p"-(1 -p). Use the Chernoff bound to show that Pipi(X) > po(X) |Ho) Ao find the best Chernoff bound on Pe,0 = P{p1 (X) > po (X) | Ho } and Pe,1 = P{po(X) 2 p1 (X) H1 }. Let s* be the optimal value of s for minimizing the bound to Pe,i. Show sf = 1 - so. (d) Again let X1, ...,X2n be independent discrete random variables with X, taking nonnega- tive integer values only. Let 2n Atie-No Po(X1, . . ., X2n) = x1 20, 1gis 2n i= xi! i=n+1 xi