Question

Q1] (20 Marks) Three-digit numbers constructed using only 0's and 1's. Let X(s) - No. of one's in the number,Y (3) - index of the first one" in the number. For the two discrete random variables, X(s) and Y (s) described in Fig. (1) Below, implement: ) The joint probability mass function Pxy(x, y), represented in a suitable table. (ii) The joint distribution function Fxx (x, y) P(X < x, Y sy), represented in a (ii) The marginal probability px. (iv) The marginal probability py. X. Fig. (1)

Answer 1

(i)

Sample space={000,001,010,100,011,101,110,111}

Pxy(0,0) = P({000}) = 1/8; Pxy(1, 1) = P({100}) = 1/8; Px,y(1, 2) = P({010}) = 1/8; Px,y(1,3) = P({001}) = 1/8; Px,y(2, 1) = P({110, 101}) = 2/8; Px,y(2, 2) = P({011}) = 1/8; Px.y(3, 1) = P({111}) = 1/8. Joint pmf of (X,Y) is given as follows : 1 Y 0 2 3 0 1/8 0 0 0 0 1/8 1/8 1/8 2 0 2/8 1/8 0 3 0 1/8 0 0 Totals 1/8 4/8 2/8 1/8 1 Total s 1/8 3/8 3/8 1/8 Х 1

ii. Joint CDF of X, Y is Fx,y(1,Y) = P(X<2, Y <y) Y 0 1 2 3 0 1/8 1/8 1/8 1/8 1 1/8 2/8 3/8 4/8 X 2 1/8 4/8 6/8 7/8 3 1/8 5/8 7/8 1 in. px(x) = 1/8 if x = 0,3 = 3/8 if x = 1,2 = 0 otherwise iv. py(y) = 1/8 if y=0,3 = 4/8 if y=1 = 2/8 if y=2 = 0 otherwise