X and Y are two independent discrete random Variables. The CDFs as follows: x 0 4 5 F(x) 0.3 0.6 1 y 1 5 7 9 F(y) 0.2 0.5 0.6 1 Z is new random variable defined as Z = 3X+4Y+2. Find the following: Moment generating function of z: A table consisting of values of z and F(z) Find the first moment of Z

Determine E(x) and E(y)

solve it hand written part 3

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