Question

Assume that the joint density function of X and Y is given by f (x, y) = 4,0 < x < 2,0 < y = 2 and f (x, y) = 0 elsewhere. (a) Find P (X < 1, Y > 1). (b) Find the joint cumulative distribution function F(x, y) of the two random variables. Include all the regions. (c) Find P (X<Y). (d) Explain how the value of P (1 < X < 2,1 < Y < 2) can be found by substitution into the func- tion F(x, y). (e) Explain why P (X<Y) cannot be found by substitution into the function F(x, y).

Answer 1

We have ourono istaga on OLE L2, OLYLL flang) elsew here.. 9 to ali P(x31, 1919- f(x,y) dy dat 1-бы , " I dy dx 4 2-0 y=1 dn 4 dn 4 10 X2 to da 4 a=0 4 P(XS1, 13, 1) = 4 4.

b) Far (94) - "Soffor (sit) at de S=0 to I dt ds . 4 3=0 to t ds SCO ag y ds 4. S=0 [syna . 4 S=0 xy Exiy (x,y) 9 OLXL2 4 (6) and oLys2 We want : P(XLY) = Ply>x)

P(XLY) 2 (2,2) SUE i exlay o 2: de dy [2 tý dy 1.0 2 y dy 4 o € P.(XLY) y272 1 P(Xey) - 8 li

(d) Fx(x) = P1Xsx) Ex,y (x, y) = P(XLX GYLY) 80we, want : PCI 4XL2 , 119.12) = P(X52, YL2) – P(x 41,4211 = Fry (2,2) - Exy (1:1) (2)(2) (1)(1) f using 4 9 १ 4 part (bij 1 4 3 (e) We cannot find the probability PLXCY) using caf (or Joint Caf) because Fx (0) PIX.CZ) Here, x Ý random Variable and se is value from this random varible.