Question

Please help me to solve this problem. Thank you.

thst Xmnd Y Lal XY be randem variobles; preve indepndent f) = giag,4 Where ake b, cxydihIR

Answer 1

TOPIC:Independence of random variables

Define the basic de dinition dons independence two sandom variables x and Y between as - It x' and y are independent; then, Exix (X,Y) = Fx (x) Gy (7) ; &n, f. > P (X<X, *%) =P(x<). p(Y4) ; *2,7 1 - where, Exiy (217) = Joint edf of (X,Y), Fx (x) = marginal cdf of x, Gy (7) = marginal cap of y.) Now, we need to prove that, X and Y are independent. (= dxix (x, y) = g, (x). J (I) ; & a<x <b, clyd. I where txiy (742) = Joint Add of (x, y, g. (x) = Marginal pdf of x; 92 (1) ,

IF PART- Given that, 8x, y (x, y) = 9, (x). J₂ (2) ; acxeb, CRYLd. we need to prove that, X and Y are independent : ie. P (xLX, Y EY) = P(x= x) . P (Y = "); xxiy. (BY - A).

=) Now, we have P (x Ex, YEY E der Ctr, tu) dt, dt g, (+1). Iz (+2) . dt, dt ₂1 11:2.00m) { f29333 = P(x2x), 8 (427) ; & xey. Thus, SP (xx, YEY) =P ( x 32). P (427) tx, y, - Ex, y (x, y) = F x (x). Gy (7) ; & X, Y =) X and Y are independent. (proved,

7 ONLY IF PART :- Criven that x and r are independent We need to prove that x,y (267) = g(x). 9, 17) ; x 79. Now, since x and y are independent, we must have, Ex, y (x, y) = FX (2). Gy (Y). [By- Now, the joint pdf of x and y can be obtained as - → dry(4) dy [foy (2, 4]. (04-) - men og brace.oce). 7 28,() = [Fx (27) and g (7) - dy [Gy( = 9(2). 22(): &x,y. Hences / dx,y (xet) a g, (x). J₂ (7) va < x <ь, ccycd. (Proved)