Question

Q#1 Let X and Y are joint probability functions given by a- f(x, y) = *y*; x = 1, 2, 3; y = 1, 2 b- f(x,y) = 5%; x = 2,4,5; y = 1, 2, 3 Find the marginal probability functions of r.v X&Y also find out if X & Y are independent? Q#2 Let X denotes the number of times a certain numerical control machine will malfunction: 1, 2, or 3times on any given day. Let Y denote the number of times a technician is called on an emergency call. Their joint probability distribution is given as х f(x, y) 1 2 3 1 0.05 0.05 0.10 Y 3 0.05 0.10 0.35 5 0.00 0.20 0.10 (a) Evaluate the marginal distribution of X. (b) Evaluate the marginal distribution of Y. (c) Find PlY = 3 / X = 2). Q#3 In testing a certain kind of truck tire over rugged terrain, it is found that 25% of the trucks fail to complete the test run without a blowout. Of the next 15trucks tested, find the probability that (a) From 3 to 6 have blowouts; (b) Fewer than 4 have blowouts; (c) More than 5 have blowouts.

Answer 1

(6) x awy have joint probability function flary) = 30 ; x=1, 2, 3; y = 1,2 So, fx/X) = [ f(qiy) y= ху2 2 y=1 Že 2 x2 30 xx, 2 30 2.22 and 3 30 V tyly) = 3 x + 42 2 41811) x=1 I nya 30, xa 1, 2, 3 x=1 30 ར 1.92 2.42 30 % 30+ 3у2 30 ; ya 1,2 Note that f (x1y) + fx(x)fy(4) X aw .y f(719) so fx(x) = such a Not indepewent. y2 5 ;y=1,2; (6) ay 66 ; x=2, 415 ; 9=1, 2, 3 . 3 I 66 y=1 2x + 66 6x 66 5 ww 2x2 2x + 6.6 66 ; 2 T ; x=2,415 64 fy(9) - Ž zy W 2 66 2y 06 + Ily 66 2 z ; y: 1,2,3. Note that f(x,y) fx (²) fy(y) X awy are indepedent. 82 f(x,y) Х 2 3 - (1) The Marginal distribution of x 0.05 0.05 0.10 0-20 is 2 3 0.35 10.50 0.05 3 fx (a) 0.10 0.35 0.55 O-00 0.20 0.10 0.30 ما۔ 0.10 0.85 0.65 1 y 3 5 (6) The marginal distribution of Y is Sy(9) 0.20 0.50 0.30 CC) 0.10 P(Y=3x=2) = P(Y30X-2) P (x=2) where PCX Z 20 Y-3) = f(218)=0.10. Plx+2) = f(2)=0-35 0.35 10 35