A certain market has both an express checkout line and a superexpress checkout l

Question

A certain market has both an express checkout line and a superexpress checkout line. Let X_1 denote the number of customers in line at the express checkout at a particular time of day, and let X_2 denote the number of customers. In line at the superexpress checkout at the same time. Suppose the joint pmf of x_1 and x_2 is a given in the accompanying table. (a) What is P(x_1 - 1, x_2 - 1), that is, the probability that there is exactly one customer in each line? P(x_1 = 1, x_2 = 1) = (b) What is P(x_1 = x_2), that is, the probability that the numbers of customers in the two lines are identical? P(X_1 = X_2) = (c) Let A denote the event that there are at least two more customers In one line than ln the other line. Express A In terms of X_1 and X_2. A = {X_1 lessthanorequalto 2 + X_2 Union X_2 greaterthanorequalto 2 + X_1} A = {X_1 greaterthanorequalto 2 + X_2 Union X_2 greaterthanorequalto 2 + X_1} A = {X_1 lessthanorequalto 2 + X_2 Union X_2 lessthanorequalto 2 + X_1} A = {X_1 greaterthanorequalto 2 + X_2 Union X_2 lessthanorequalto 2 + X_1} Calculate the probability of this event. P(A) = (d) What is the probability that the total number of customers in the two lines is exactly four? At least four? P(exactly four) = P(at least four) =

Explanation / Answer

(a)

P(X1=1, X2=1) = 0.15

(b)

P(X1=X2) = P(X1=0, X2=0)+P(X1=1, X2=1)+P(X1=2, X2=2)+P(X1=3, X2=3) = 0.08+0.15+0.10+0.07 = 0.4

(c)

Second option is correct.

Following table shows the values of X1 and X2 for which X2 >= X1+2:

Following table shows the values of X1 and X2 for which X1 >= X2+2:

So requried probability is 0.08+ 0.14 =0.22

(D)

Folloiwng table shows the X1 and X2 where X1+X2=4:

P(exactly four) = 0.16

Folloiwng table shows the X1 and X2 where X1+X2> =4:

P(at least 4) = 0.48

X1 X2 P(X1=x1,X2=x2) 0 2 0.04 0 3 0 1 3 0.04 Total 0.08

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