Question

. An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X represent the number of points earned on the first part and let Y represent the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the table below. Y 0 5 10 15 0 0.11 0.07 0.05 0.12 X 5 0.07 0.14 0.09 0.07 10 0.06 0.10 0.04 0.08 (a) (3Calculate P(X ≥ Y ). (b) (4 points) Find the conditional distribution of P(Y |X = 10)

Answer 1

PMF of X and Y

P(x,y)		X
Y	0	5	10
0	0.11	0.07	0.06
5	0.07	0.14	0.1
10	0.05	0.09	0.04
15	0.12	0.07	0.08

(a) P(XY)

XY is true only for the following events

For X = 0 ; Y=0 ; P(X=0,Y=0) = 0.11

For X=5; Y=0 ; P(X=5,Y=0) = 0.07 ; X=5, Y=5 ; P(X=5,Y=5) = 0.14

For X=10, Y=0 ; P(X=10,Y=0) = 0.06 ; X=10, Y=5 ; P(X=10,Y=5) = 0.1 ; P(X=10,Y=10) = 0.04

P(XY) = P(X=0,Y=0) + P(X=5,Y=0) + P(X=5,Y=5) + P(X=10,Y=0)+P(X=10,Y=5)+P(X=10,Y=10)

= 0.11+0.07+0.14+0.06+0.1+0.04 = 0.52

P(XY) = 0.52

(b) Conditional Distribution of P(Y|X=10)

PX(10) : Marginal Distribution of X for X=10

For Y=0;

For Y=5

For Y=10

For Y=15

Y	P(Y|X=10)
0	0.214285714
5	0.357142857
10	0.142857143
15	0.285714286