Answer Happy

4) Properties of Joint Distribution Functions (Discrete or Continuous): a) FxY(-00,-) = 0 Example Given: Fxx(x,y) = 0.2u(x - 1)u(y - 1) + 0.3u(x - 2)u(y - 1) + 0.5u(x – 3)u(y - 3) Then: Fx.y(-0,-) b) Fx.r(0,0) = 1 Example Given: Fxy(x,y) = 0.2u(x - 1)u(y - 1) + 0.3u(x - 2)uly - 1) + 0.5u(x – 3)u(y - 3) Then Fx.x(+00, +00) C) OS Fxy(x,y) S1 Proof: Fxx(-0,-) = 0 since no joint-outcomes occur at these values (with any probability). Fxx(0,0) = 1 since all joint-outcomes will occur at or below these values. Since there are no negative probabilities, all other joint distribution function values fall in between the extremes of O and 1. d) Fxx(x,y)is a nondecreasing function of both x and y Proof: Since all probabilities are non-negative, cumulative probabilities cannot decrease. e) P(x, < X 5 X2,Y1 <Y S y2) = Fx.x(x2.Y2) - Fx.r(xx,y) - FxY(X1,Y2) + Fx.r(x,y) The 4th term adds-back one of the two copies of the lower left area subtracted by the 2nd and 3* terms. See diagram. (x1.72) (x2.72) I. 0.5 (x,y) (Xzy) 0