Question

l is I unit and that of Question 3: Two machines produce a certain item. The capacity per day or N Machine-2 is 2 units Let A, , be the discrete random variables for the actual daily production o i and 2, respectively. The joint probubilit y mass function is given below 0.0s o.10 0.0s 0.20 0.50 010 a. (10 points) Are X and X, independent? Justify your answer b. (10 points) Write the conditional probability mass function of Xi given that X2 = 1. (10 points) Write the probability mass function of X-X

Answer 1

P(X1 = 0 , X2 = 1) = 0.1

P(X1 = 0) = 0.05 + 0.1 + 0.05 = 0.2

P(X2 = 1) = 0.1+0.5=0.6

P(X1 = 0) * P(X2 = 1) = 0.2 * 0.6 = 0.12

since

P(X1 = 0 , X2 = 1)  $\neq$ P(X1 = 0) * P(X2 = 1)

X1 and X2 are not independent

b)

P(X1 = x|X2 = 1) = P(X1 = x and X2 = 1)/ P(X2 = 1)

P(X1 = 0|X2 = 1) = 0.1/0.6 = 1/6

P(X1 = 1 |X2 = 1) = 0.5/0.6=5/6

c)

x1	x2	x1-x2	p
0	0	0	0.05
0	1	-1	0.1
0	2	-2	0.05
1	0	1	0.2
1	1	0	0.5
1	2	-1	0.1
	x1-x2	p
	-2	0.05
	-1	0.2
	0	0.55
	1	0.2