Question

A certain market has both an express checkout line and a superexpress checkout line. Let X? denote the number of customers in line at the express checkout at a particular time of day, and let X? denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of X? and X? is as given in the accompanying table. X? 0 1 2 3 0 0.08 0.06 0.04 0.00 1 0.06 0.15 0.04 0.04 X? 2 0.05 0.04 0.10 0.06 3 0.01 0.02 0.04 0.07 4 0.00 0.01 0.05 0.08 (a) What is P(X? = 1, X? = 1), that is, the probability that there is exactly one customer in each line? P(X? = 1, X? = 1) = (b) What is P(X? = X?), that is, the probability that the numbers of customers in the two lines are identical? P(X? = X?) = (c) Let A denote the event that there are at least two more customers in one line than the other line. Express A in terms of X? and X?. A = {X? ? 2 + X? ? X? ? 2 + X?} A = {X? ? 2 + X? ? X? ? 2 + X?} A = {X? ? 2 + X? ? X? ? 2 + X?} A = {X? ? 2 + X? ? X? ? 2 + X?} 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) =

          A certain market has both an express checkout line and a superexpress checkout line. Let X? denote the number of customers in line at the express checkout at a particular time of day, and let X? denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of X? and X? is as given in the accompanying table.

X?
	0	1	2	3
0	0.08	0.06	0.04	0.00
1	0.06	0.15	0.04	0.04
X?	2	0.05	0.04	0.10	0.06
3	0.01	0.02	0.04	0.07
4	0.00	0.01	0.05	0.08

(a) What is P(X? = 1, X? = 1), that is, the probability that there is exactly one customer in each line?
P(X? = 1, X? = 1) = 

(b) What is P(X? = X?), that is, the probability that the numbers of customers in the two lines are identical?
P(X? = X?) = 

(c) Let A denote the event that there are at least two more customers in one line than the other line. Express A in terms of X? and X?.
A = {X? ? 2 + X? ? X? ? 2 + X?}
A = {X? ? 2 + X? ? X? ? 2 + X?}
A = {X? ? 2 + X? ? X? ? 2 + X?}
A = {X? ? 2 + X? ? X? ? 2 + X?}

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) =

A certain market has both an express checkout line and a superexpress checkout line. Let X? denote the number of customers in line at the express checkout at a particular time of day, and let X? denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of X? and X? is as given in the accompanying table.

X?
0 1 2 3
0 0.08 0.06 0.04 0.00
1 0.06 0.15 0.04 0.04
X? 2 0.05 0.04 0.10 0.06
3 0.01 0.02 0.04 0.07
4 0.00 0.01 0.05 0.08

(a) What is P(X? = 1, X? = 1), that is, the probability that there is exactly one customer in each line?
P(X? = 1, X? = 1) =

(b) What is P(X? = X?), that is, the probability that the numbers of customers in the two lines are identical?
P(X? = X?) =

(c) Let A denote the event that there are at least two more customers in one line than the other line. Express A in terms of X? and X?.
A = X? ? 2 + X? ? X? ? 2 + X?
A = X? ? 2 + X? ? X? ? 2 + X?
A = X? ? 2 + X? ? X? ? 2 + X?
A = X? ? 2 + X? ? X? ? 2 + X?

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) =

Added by Miguel S.

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