Question

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I. Over the past 300 days, DiCarlo has experienced 54 days with no automobiles sold, 117 days with 1 automobile sold, 72 days with 2 automobiles sold, 42 days with 3 automobiles sold, 12 days with 4 automobiles sold, and 3 days with 5 automobiles sold. Suppose we consider the experiment of observing a day of operations at DiCarlo Motors and define the random variable of interest as x the number of automobiles sold during a day. a) Use the relative frequency method to develop a probability distribution for the number of cars sold per day at DiCarlo Motors in Saratoga, New York. b) The expected value for the number of automobiles sold during a day c) The variance for the number of automobiles sold during a day

Answer 1

Solution

Given x = number of automobiles sold during a day, the given frequency and the related requisite calculations are shown in the following table.

i	Number of automobiles sold during the day (xi)	Number of days (Frequency) (fi)	Relative Frequency Ri = p(xi) = fi/300	xi.p(xi)	xi2.p(xi)
1	0	54	0.18	0.00	0.00
2	1	117	0.39	0.39	0.39
3	2	72	0.24	0.48	0.96
4	3	42	0.14	0.42	1.26
5	4	12	0.04	0.16	0.64
6	5	3	0.01	0.05	0.25
	Total	300	1.00	1.50	3.50

Part (a)

Probability Distribution

Number of automobiles sold during the day (xi)	Probability
0	0.18
1	0.39
2	0.24
3	0.14
4	0.04
5	0.01
Total	1.00

ANSWER

Part (b)

Expected number of cars sold per day = E(X) = ∑[i = 1, 6]{xi.p(xi)} = 1.50 ANSWER [refer to column 5 of table at the top]

Part (c)

Variance of the number of cars sold per day = V(X)

= E(X²) – { E(X)}²

= ∑[i = 1, 6]{xi².p(xi)} – 1.5² [refer to the answer of Part (b)]

= 3.5 – 2.25 [refer to column 6 of table at the top]

= 1.25 ANSWER