Question

Let X denote the number of times (1, 2, or 3 times) a certain machine malfunctions on any given day.

2. Let X denote the number of times (1, 2, or 3 times) a certain machine malfunctions on any given day. Let Y denote the number of times (1, 2, or 3 times) a technician is called on an emergency call. The joint probability distribution fxy(x, y) is given by 1 2 y 0.05 1 0.05 0.1 2 0.05 0.1 0.35 0 0.2 0.1 (a) Evaluate the marginal pdf and the mean of X (b) Evaluate the marginal pdf and the mean of Y (c) Evaluate P(Y = 3|X 2) (d) Evaluate E(XY) and determine if X and Y are uncorrelated.

Answer 1

(a) x takes values 1,2,3. Then, 0. (X ) = 0.05 +0.05 +0 - P(x - 2) = 0:05+0,1 +0.2=0-35 PLX = 3) 0.1t 0-35+ 0.13 0.55 P.M.F of x = P(x=2) The 0.1 when ! - 0.35 when x=2 = 0.55 when x=3

Mean of x = E() = 3 2 P(x= x) 221 = X 0+ 2 * 0.35 + 3X0.55 -.0.1+ 0.7+ 1.65 : 2.45 (6) y takes values 19213 Then, P(y=1) = 0.05 +0.05 +0.1= 0.2 P(Y=2) = 0.05 +0.1 +0.35 = 0.5 P(Y=3) = 0 + 0.2 +0.1 = 0.3

The s. Mor of 4 =P (4=4) 0.5 if y=2 0.3 if y=3

Man of y = E(4) y Ply=y) = 10 2 + 2x0.5 + 3 0:3 © PY = 31= 2) PCY=3, *22) P(x=2) 0-2 0.35 = 0.571 a) E( XY) 3 22 3 4 7 (x = x 22141 3 ) -1.1.0.05 + 1x 20.05130 + 2xIxo.05+ 2x2x o.1 + 2x3x0.2 +1x3* 0.1 +283 x 0.35 +x3 0.1 -0.05 +0.1 +0.1 +0.4 +1.2 +043 +2.1 + 0.9 5.15

X and Y no + are correlated We then know that if cov (x40 Here, COV( x4) = E(XY) - E( E(Y) = 5:15 - (2045) * 2.1 - 5.15 5.145 But I 0.005 here con(X,Y)= 0.005 to X and y arte correlated So, here.