3.40
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Let Y denote a random variable that has a geometric distribution, with a probability of success on any trial denoted by p.
a) Find P(Y>=2) if p=0.1
Ans: 1-P(Y=1) = 0.9 ,,,,,,, I understand part a). I got correct ans.
b) Find P(Y>4 | Y>2) for general p. Compare this result with the unconditional probability P(Y>=2).[This property is referred to
as "lack of memory"]
Ans: Since Y>4 is a subset of Y>2, i thought,
P(Y>2)= 1-P(Y=2) = 1-(1-P)*P but answer is wrong
correct ans= (1-p)^2 , why? maybe I wasn't suppose to use
compliment.
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3.54
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Let Y denote a random variable that has a POISON distribution with mean
λ =2. Find the following probabilities.
Table λ x equals
------------------------------------
2 0 .135
2 1 .406
2 2 .677
2 3 .857
2 4 .947
a) P(Y=4)
Ans: I understand this one, if i didn't i would be in deeper trouble :-)
b) P(Y>=4)
Ans: I thought 1-P(y=4)- P(y=3) - P(y=2) - P(y=1) = 0.187 ,,
incorrect ans.
correct ans: 0.143
So, I played around with it a while and I found that it's : 1-
P(Y=3): but why. wouldn't you have to subtract y=2 and 1 as well?
c) P(Y<4)
why is correct answer, only P(X=3)?
d) P(Y>=4 | Y>=2)
I thought about using conditional probability, would that be correct to use?
Correct Ans: 0.2407
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3.58
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In the article cited in Exercise 3.57
( the national maximum speed limit NMSL of 55 miles per hour was imposed in the U.S. in early 1974. The benefits of this law have been studied by D.B. Kamerud (1983), who reported that the fatality rate for interstate highways with the NMSL in 1975 was apporximately 16 per 10^9 vehicle miles.),
the projected fatality rate for 1975 if the NMSL had not been in effect was 25 per 10^9 vehicle miles. Assume that these conditions had prevailed.
a) Find the probability that at most 15 fatalities occurred in a given block of 10^9 vehicle miles.
b) Find the probability that at least 20 fatalities occurred in a given block of 10^9 vehicle miles.
ANS: not sure on what distrubution to use.
Correct Ans:
a) 0.022
b) 0.866
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3.64
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The number of imperfections in the weave of a certain textile has a Poisson distribution with a mean of four per square yard.
a) Find the prob. that a 1-square-yard sample will contain at least one imperfection.
Ans: I understand
1-P(Y=0) = 0.982
b)Find the probability that a 3-square-yard sample will contain at least one imperfection.
Ans: not sure what to do.
Correct Ans: 1- e^(-12)
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3.66
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The number of bacteria colonies of a certain type in samples of polluted water has a Poisson distribution with a mean of two per cubic centimeter.
a)If four 1-cubic-centimeter samples of this water are independently selected, find the probability that at least one sample will contain one or more bacteria colonies.
b)How many 1-cubic-centimeter samples should be selected to establish a probability of approximately 0.95 of containing at least one bacteria colony?
Ans: not sure how to set-up and do problem.
Correct Ans:
a) 0.9997
b)2
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3.69
The number of cars entering a parking lot is a random variable that has a Poisson distribution, with a mean of four per hour. The lot holds only twelve cars.
a)Find the probability that the lot will fill up in the first hour. (Assume that all cars stay in the lot longer than one hour)
b)Find the probability that fewer than twelve cars will arrive during an eight-hour day.
Ans: It's a good thing the problem tells you what distribution it has but still I have problems setting up problem. I would think P(Y=1) but it's not that simple since i got answer wrong.
Correct Ans:
a) 0.001
b) 0.000017