Question

Linda is a sales associate at a large auto dealership. At her commission rate of 25% of gross profit on each vehicle she sells, Linda expects to earn 360 for each car sold and 410 for each truck or SUV sold. Linda motivates herself by using probability estimates of her sales. For a sunny Saturday in April, she estimates her car sales as follows:

Cars sold	0	1	2	3
Probability	0.3	0.35	0.15	k

Linda's estimate of her truck or SUV sales is

Truck or SUV sold	0	1	2
Probability	0.4	0.45	0.15

Calculate:

5. Lindas best estimate of her earnings for the day.

6. the variance of the number of cars Linda sells for the day.

7. the variance of the number of trucks or SUVs Linda sells for the day.

8. the variance of Linda's earnings for the day.

Answer 1

5)

The joint distribution total number of vehicle sold and earning is obtained as follows

Probability = P(# cars sold)*P(# trucks/SUV sold)

Profit = (# cars sold)*360 + (# trucks/SUV sold)*410

Cars sold	Truck/SUV sold	Prob	Earning
0	0	0.12	0
0	1	0.135	410
0	2	0.045	820
1	0	0.14	360
1	1	0.1575	770
1	2	0.0525	1180
2	0	0.06	720
2	1	0.0675	1130
2	2	0.0225	1540
3	0	0.08	1080
3	1	0.09	1490
3	2	0.03	1900

The expected value for the earning is obtained using the following formula,

From the above table,

earning, X	Prob, P(X)	X*P(X)
0	0.12	0
410	0.135	55.35
820	0.045	36.9
360	0.14	50.4
770	0.1575	121.275
1180	0.0525	61.95
720	0.06	43.2
1130	0.0675	76.275
1540	0.0225	34.65
1080	0.08	86.4
1490	0.09	134.1
1900	0.03	57
	Sum	757.5

6)

The variance for the number of cars sold is obtained using the following formula,

From the data table,

Cars sold, x	Prob, p	X*P(X)	X^2P(X)
0	0.3	0	0
1	0.35	0.35	0.35
2	0.15	0.3	0.6
3	0.2	0.6	1.8
	Sum	1.25	2.75

7)

From the data table,

Truck/SUV sold	Prob	X*P(X)	X^2P(X)
0	0.4	0	0
1	0.45	0.45	0.45
2	0.15	0.3	0.6
	Sum	0.75	1.05

8)

Earning, X	Prob, P(X)	X*P(X)	X^2P(X)
0	0.12	0	0
410	0.135	55.35	22693.5
820	0.045	36.9	30258
360	0.14	50.4	18144
770	0.1575	121.275	93381.75
1180	0.0525	61.95	73101
720	0.06	43.2	31104
1130	0.0675	76.275	86190.75
1540	0.0225	34.65	53361
1080	0.08	86.4	93312
1490	0.09	134.1	199809
1900	0.03	57	108300
	Sum	757.5	809655