(1) QNB branch at QU has two cashier lines, one is known to be faster than the o
ID: 3321260 • Letter: #
Question
(1) QNB branch at QU has two cashier lines, one is known to be faster than the other. Let X denote the number of customers in the first line and X2 denote the number of customers in the second (fast) line at the same time of the day. Suppose that the joint p.m.f. ofXi and X2 is given by the shown table. (a) What is the probability that both lines X1 are empty? b) What is the probability that both lines T 2 0 0.05 00 0 have the same length (i.e. same number of customers)? 1 0.15 0.05 00 X2 2 0.1 0.150 0 (c) What is the probability that the total 3 0.05 0.05 0.1 0 4 0.05 0.05 0.15 0.05 number of customers in the two lines is exactly four? (d) What is the probability that the faster line has more than 1 customer given that the (e) Find the correlation between Xi and X2 and comment on the existence and strength of (f) Are Xi and X2 independent? Why? first line is empty? linear relation between Xi and X2.Explanation / Answer
(a)
P(Both lines are empty)
= P(X1 = 0 and X2 = 0)
= 0.05
(b)
P(both lines have same length)
= P(X1 = X2)
= P(X1 = X2 = 0) + P(X1 = X2 = 1) + P(X1 = X2 = 2) + P(X1 = X2 = 3)
= 0.05 + 0.05 + 0 + 0
= 0.1
(c)
P(total no. Of customers in 2 lines = 4)
= P(X1 + X2 = 4)
= P(X1 = 0, X2 = 4) + P(X1 = 1, X2 = 3) + P(X1 = 2, X2 = 2) + P(X1 = 3, X2 = 1)
= 0.05 + 0.05 + 0 + 0
= 0.1
(d)
P(faster line has more than 1 customer given that first line is empty)
= P(X2 > 1 | X1 = 0)
= P(X2 > 1 & X1 = 0) / P(X1 = 0)
= (0.1 + 0.05 + 0.05) / (0.05 + 0.15 + 0.1 + 0.05 + 0.05)
= 0.2 / 0.4
= 0.5
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