Question

Subject: Limit theorem 1. The number of cars sold weekly at a dealership is a random variable with expected value 16. Provide an upper limit on the probability that a) Sales made in the next week exceed 18; b) Suppose that a variation in the number of cars sold weekly is equal to 9. Provide a lower limit for a probability of a close number of sales week is between 10 and 22, inclusive.

Answer #1

**Solution**
**:**

**Let the random variable X represents number of cars sold
weekly at a certain delarship .**

**Given that expected value (mean value) of the random
variable X is 16**

**If the random variable X takes only nonnegative values
then for any value a>0**

(a) **Then the probability that the next week sales exceed
18 is calculated as follows **

**= 16/19**

**= 0.8421**

**(b) P(10<X<22) =
**

**= P(- 2 < z < 2) = P(z < 2) - P(z
< - 2)**

**= 0.9772 - 0.0228**

**= 0.9544**

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weeks.
Week
Sales (1,000s
of gallons)
1
17
2
21
3
19
4
23
5
18
6
16
7
20
8
18
9
22
10
20
11
15
12
22
(a)
Compute four-week and five-week moving averages for the time
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Time Series
Value
4-Week
Moving
Average
Forecast
5-Week
Moving
Average
Forecast
1
17
2
21
3
19...

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20
0
28
34
18
35
14
−21
−18
−16
y:
22
−3
25
23
14
8
16
−8
−11
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Do bonds reduce the overall risk of an investment portfolio? Let
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38
0
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24
19
16
36
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27
25
16
15
15
−10
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Do bonds reduce the overall risk of an investment portfolio? Let
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Do bonds reduce the overall risk of an investment portfolio? Let
x be a random variable representing annual percent return for
Vanguard Total Stock Index (all stocks). Let y be a random variable
representing annual return for Vanguard Balanced Index (60% stock
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