Exercise 32.12. Vending machine. The time between consecutive uses of a vending
ID: 2934169 • Letter: E
Question
Exercise 32.12. Vending machine. The time between consecutive uses of a vending machine is Exponential with average 15 minutes a. Given that the machine has not been used in the previous 5 minutes, what is the probability that the machine will not be used during the next 10 minutes? b. How many purchases are expected within the next hour? c. What is the distribution of the number of purchases to be made within the next hour? d. Suppose that a person pays 75 cents for each beverage in the vending machine. The supplier pays 40 cents per beverage in the machine, and thus makes a profit of 35 cents per beverage. What is the expected profit made during a given 8 hour period?Explanation / Answer
Ans:
a)As,exponential distribution is memoryless:
P(x>10)=e-10/15=0.5134
b)Expected number of purchases within next hr=60/15=4 per hr.
c)Number of purchases in next hr will follow poisson distribution with mean=4 per hr.
d)8*4=32 per 8 hr
caculate POISSON(k,32,FALSE) and profit=k*0.35*POISSON(k,32,FALSE)
Add up all profit columns.
Expected profit=11.2
k Poisson profit k Poisson profit 0 0.0000 0.0000 31 0.0703 0.7632 1 0.0000 0.0000 32 0.0703 0.7878 2 0.0000 0.0000 33 0.0682 0.7878 3 0.0000 0.0000 34 0.0642 0.7639 4 0.0000 0.0000 35 0.0587 0.7190 5 0.0000 0.0000 36 0.0522 0.6574 6 0.0000 0.0000 37 0.0451 0.5843 7 0.0000 0.0000 38 0.0380 0.5054 8 0.0000 0.0000 39 0.0312 0.4256 9 0.0000 0.0000 40 0.0249 0.3492 10 0.0000 0.0000 41 0.0195 0.2793 11 0.0000 0.0000 42 0.0148 0.2180 12 0.0000 0.0001 43 0.0110 0.1661 13 0.0001 0.0003 44 0.0080 0.1236 14 0.0002 0.0008 45 0.0057 0.0899 15 0.0004 0.0019 46 0.0040 0.0639 16 0.0007 0.0041 47 0.0027 0.0445 17 0.0014 0.0082 48 0.0018 0.0303 18 0.0024 0.0154 49 0.0012 0.0202 19 0.0041 0.0274 50 0.0008 0.0132 20 0.0066 0.0462 51 0.0005 0.0084 21 0.0101 0.0739 52 0.0003 0.0053 22 0.0146 0.1126 53 0.0002 0.0033 23 0.0203 0.1638 54 0.0001 0.0020 24 0.0271 0.2279 55 0.0001 0.0012 25 0.0347 0.3039 56 0.0000 0.0007 26 0.0427 0.3890 57 0.0000 0.0004 27 0.0507 0.4787 58 0.0000 0.0002 28 0.0579 0.5674 59 0.0000 0.0001 29 0.0639 0.6484 60 0.0000 0.0001 30 0.0681 0.7155Related Questions
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