During a total of 60 days, daily sales records for a car dealership show that it

Question

During a total of 60 days, daily sales records for a car dealership show that it may sell 0,1,2, or 3 cars. The following table gives the number of days on which 0, 1, 2, or 3 cars were sold, respectively. Let X be the number of cars sold on a random day. Below is the pmf of X:

Number of cars sold 0 1 2 3 Number of days 30 18 9 3

Randomly pick up a specific day out of these 60 days and let X denote the number of cars sold on that day. Answer the following questions. (For each question you must show your calculation steps to earn credits. Any answer with no work will receive zero credit.)

(1) (4 points) Calculate the pmf of X.

(2) (2 points) Calculate the probability that more than one car is sold on

this selected day.

(3) (2 points) Calculate the expected value of X.

(4) (4 points) Calculate the variance of X.

Explanation / Answer

Total number of days = 30 + 18 + 9 + 3 = 60 days

Thus, the pmf would be:

P (X = 0) = 30 / 60 = 0.5

P (X = 1) = 18 / 60 = 0.3

P (X = 2) = 9 / 60 = 0.15

P (X = 3) = 3 / 60 = 0.05

Thus, P (X > 1) = 0.05 + 0.15

= 0.20

E(X) = 0 (0.5) + 1(0.3) + 2(0.15) + 3(0.05)

= 0 + 0.3 + 0.3 + 0.15

= 0.75 cars / day

V(X) = E(X2) - [E(X)]2

= [ 02(0.5) + 12(0.3) + 22(0.15) + 32(0.05)] - (0.75)2

= [ 0 + 0.3 + 0.6 + 0.45] - 0.5625

= 1.35 - 0.5625

= 0.7875

Hope this helps.

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