7. Let X₁, X2,..., Xn be independent variables, X; being N(µi, 1), and let y = = x² + x² +...+X²/2. Show that the characteristic function of Y is by(t) = 1 (1-2it)n/2 exp ite 1-2it where 0 = ²+²+ ...+. The random variables Y is said to have the non-central chi-squared distribution with n degrees of freedom and non-centrality parameter 0, written x² (n; 0).

Suppose that the random variables Y1,..., Y(n > 2) satisfy Y - Ba, + Ei, i= 1,2,..., T, where 1,..., En are iid N(0, o), and both B and o are unknown. (a) Assume z1,..., In are fixed known constants. Here we observe Y1 = n,., Y observed data Y = y = (y1,..., Yn). Find a two-dimensional sufficient statistic of Y = (Yı,., Yn) for (8,0?). (b) Assume now that a1,...,n are random variables with a known joint distribution m(r1,..., In), and the r,'s are independent of G's (it is traditional in the linear regression to use lower case for independent variables r;'s). In this case, the observed data (Y, x) = {(Yi, r;)}i-1,n. Find a three- dimensional sufficient statistic of (Y, x) for (B,02). Im, and the