My main problem is deciding with discrete distribution to use: BERNOULLI, BINOMIAL, DISCRETE UNIFORM, GEOMETRIC NEGATIVE BINOMIAL, OR POISSON. Every time I choose one, it's the wrong one. Is there some way I can easily find out which one to use. Because what I do now is I choose by trial and error ,which takes me a long time but I get the work done sometimes. When it says: the probability that at most or at least sometimes gets me confused because I can sometimes set it up like
P(y>=2) but I don't know what to do from there, use the compliment
or not. Can you please explain how you came to the decision of choosing the discrete distribution you chose. i would greatly appreciate it.
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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
ë =2. Find the following probabilities.
Table ë x equals
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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
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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
When you ask his many questions in a single post, with no evidence of your wown thought process, your chances of getting help here are greatly diminished.
I already helped you with #3.66 part b. For part a, the chances of getting zero cultures in a single 1 cm^3 sample are, from Poisson's formula, e^-2 = 0.1353. The chances of getting 0 four times in a row is 0.0003. The probability of geting at least one is therefore 0.9997
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.
Choosing the appropriate discrete distribution can be challenging, but there are some guidelines you can follow. Here's a breakdown of the different distributions and how to choose the right one:
1. Bernoulli Distribution: This distribution models a single trial with two outcomes - success (1) or failure (0), each with a constant probability p. Use this distribution when you have a binary outcome, like flipping a coin or whether a customer will buy a product.
2. Binomial Distribution: This distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p. Use this distribution when you have a fixed number of trials and want to know the probability of obtaining a certain number of successes, like the number of defective items in a production batch.
3. Discrete Uniform Distribution: This distribution models outcomes with equal probabilities. Use this distribution when each outcome has the same probability and there is no bias or preference towards any particular outcome.
4. Geometric Distribution: This distribution models the number of trials needed until the first success occurs, with a constant probability of success p. Use this distribution when you want to know the probability of the first success happening on a specific trial, like the number of attempts needed to get a specific coupon from a box of cereal.
5. Negative Binomial Distribution: This distribution models the number of trials needed to obtain a fixed number of successes, with a constant probability of success p. Use this distribution when you want to know the probability of obtaining a certain number of successes before reaching a fixed number of failures, like the number of attempts needed to find 3 flawed items in a box.
6. Poisson Distribution: This distribution models the number of events that occur in a fixed interval of time or space when these events occur with a known average rate. Use this distribution when you want to know the probability of a certain number of events occurring in a given time period, like the number of phone calls received per hour.
To choose the right distribution, you need to consider the specifics of your problem. Look for clues such as the number of trials, the probability of success, and the nature of the event you're modeling. Additionally, consider the characteristics of each distribution and how well they fit the data you have.
Now, let's go through each of the problems you provided and explain the correct distribution and reasoning behind the solutions:
Problem 3.40:
a) The problem states that Y follows a geometric distribution, which is appropriate for modeling the number of trials needed until the first success. In this case, we want to find P(Y >= 2), which is the probability that the first success occurs on the second or later trial. Since Y is defined as the number of trials until the first success, we can calculate P(Y >= 2) as 1 - P(Y = 1), which equals 0.9.
b) The problem asks for P(Y > 4 | Y > 2), which is the probability that Y is greater than 4 given that it is greater than 2. Since Y > 4 is a subset of Y > 2, we can simplify the calculation by considering the probability of Y > 4 directly. This can be calculated as (1 - p)^2, where p is the probability of success on any trial. The result is (1 - p)^2 because the first successful trial cannot be the third or fourth trial.