3. Problem 3. A given pizzeria has a dual service system: pick up and delivery.

Question

3. Problem 3. A given pizzeria has a dual service system: pick up and delivery. Let X denote the number of costumers that ask for delivery at a particular time of the day, and let Y denote the number of costumers that ask for pick up. Suppose that the joint probability mass function of X and Y is given by, 0 0.07 004 0.00 0.08 1 0.15 00 0.04 0.06 2 0.04 0.10 0.6 0.0 3 0.03 0.00 0.04 0.07 4 006 0.05 0.01 0.00 a) Caleuliate the expected difference yet (o, 1,2,3), respectively mass function px () mass function py () b) Summing ower the ridlogree of frendom, find the margiual prolwaldlity c) Summing owr the y-edegree of fredous, fiud the margiual probability

Explanation / Answer

(a) E[X -Y] = (0-0) * 0.07 + (0 - 1) * 0.04 + (0 - 2) * 0.00 + (0-3) * 0.08 + (1 -0) * 0.15 + (1 -1) * 0.05 + (1 - 2) * 0.04 + (1 - 3) * 0.06 + (2 - 0) * 0.04 + (2 - 1) * 0.10 + (2 - 2) * 0.06 + (2 - 3) * 0.05 + (3 - 0) * 0.03 + (3 - 1) * 0.00 + (3 - 2) * 0.04 + (3 - 3) * 0.07 + (4 - 0) * 0.06 + (4 - 1) * 0.05 + (4 - 2) * 0.01 + (4 - 3) * 0.00

E[X - Y] = 0.38

(b) Here px(x) = (0.07 + 0.04 + 0.08) = 0.19 ; x = 0

= (0.15 + 0.05 + 0.04 + 0.06) = 0.30 ; x = 1

= (0.04 + 0.10 + 0.06 + 0.05) = 0.25 ; x = 2

= (0.03 + 0.04 + 0.07) = 0.14 ; x = 3

= (0.06 + 0.05 + 0.01) = 0.12 ; x = 4

(c) p(y) = (0.07 + 0.15 + 0.04 + 0.03 + 0.06) = 0.35 ; y = 0

= (0.04 + 0.05 + 0.10 + 0.05) = 0.24 ; y = 1

= 0.04 + 0.06 + 0.04 + 001) = 0.15 ; y = 2

= (0.08 + 0.06 + 0.05 + 0.07) = 0.26 ; y = 3

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