Suppose that Z₁, Z2,..., Zn are statistically independent random variables. Define Y as the sum of squares of these random variables: Y = 72 i=1 Z² (n ≥2) (b) Determine My(t) for the special case that Zi~ N(0, 1).

The random variables ξ, ξ1, ξ2, . . . are independent and identically distributed with distribution P (ξ = 0) = 1/4 and P (ξ = j) = c/j for j = 1, 2, Let X0 = 0 and Xn = max(ξ1, . . . , ξn) for n = 1, 2, . . .. What value must c take? Explain why {Xn, n = 0, 1, 2,..... } is a Markov Write down the transition Draw the transition diagram and classify the states (aperiodic, transient, re- current, eorgodic, etc). Calculate P (Xn = 0). Calculate P (X4 = 3, X2 = 1|X1 = 3).

For the random variables X,Y Cov(X,Y) = -0.9 if Z=3-X then what is Cov(Z,Y)=???

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