STAT 401 Week 3 Lab 1
Geoffrey Thompson
6/3/2013
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Example Problems
There are probably more example problems than we will need, but it is good to
have them on hand to think about.
The first problem is about binomial and hypergeometric distributions, the
second problem is about calculating the expectation and variance of random
variables, and the third problem is about continuous uniform and normal distributions.
1.1
Zombie Outbreak
Unfortunately, there has been an outbreak of zombie-ism or something like it in
the country. Everybody in the city has an independent probability p of being
infected. Everybody who is infected becomes a zombie very quickly, so we will
simply refer to infected people as zombies even if they are not yet presenting
with symptoms.
This is horribly inconvenient, but we will have lab today anyway.
1. There are 50000 people in Ames and p = 0.04. If we are interested in the
number of zombies in town, what probability distribution is this? If you
are only interested in the probability that one specific person is a zombie,
what probability distribution is this?
2. Given the numbers in (1), what is the expected number of zombies in
Ames?
3. What is the variance of the number of zombies in Ames?
4. There are 40 students in this class. What is the expected number of
zombies in this class? What is the variance of the number of zombies in
this class?
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5. Suppose the class breaks into small groups to survive the zombie apocalypse. Suppose, for these questions, p = 0.1.
(a) In a group of two people, what is the probability that none are zombies? What is the probability that 1 is a zombie?
(b) What about in a group of 5 people?
(c) What about in a group of 10 people?
6. Suppose we know that 5 people in class are zombies, but we do not know
which.
(a) In a group of two people, what is the probability that 1 is a zombie?
HINT: hypergeometric.
(b) In a group of 5 people, what is the probability that all 5 are zombies?
(c) In a group of 5 people, what is the probability that exactly 2 people
are zombies?
(d) In a group of 5 people, what is the probability that none are zombies?
(e) In a group of 10 people, what is the probability that all 10 are zombies?
(f) In a group of 10 people, what is the probability that exactly 2 are
zombies?
1.2
Rental Property
Mr. Brocklehurst owns three rental properties. All three have leases expiring in
July and he still has not found new tenants. However, he knows that, for each
property, he has a probability p = 0.8 of finding a new tenant for July.
For each property, he has a fixed cost of $475.
For each property that he rents out, he receives a rent of $750.
He receives $0 for each property that he does not rent out.
For each property he does not rent out, he has an additional maintenance
expense of $50.
While it may be possible to solve the following in JMP, it is easier to do it
by hand using this table or in Excel by making a similar table. Instructions for
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filling out the table are below the table in the questions. I also hope filling out
the table will give you a better understanding of the mechanics of what is going
on with random variables and how you calculate expectations of them.
x=0
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2
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Total
P (X = x)
Profit(x)
P (x) · P rof it(x)
P (x) · P rof it(x)2
—
1. What is the expected number of rented properties?
2. In the table above, X is the number of rented properties. Fill out P (x).
3. Profit is a random variable that is a function of X. Profit(x) denotes
the profit at a particular value of x of X. Calculate the profit for each
scenario.
4. After calculating the profit for each scenario, multiply the profits by the
probability of that outcome. Sum them up to get the expected profit in
the last column.
5. Fill in the bottom row and calculate V ar(P rof it).
1.3
Continuous Zombie Problems
More bad news: there has been another zombie outbreak and it somehow involves continuous probability distributions.
1. The number of zombies in Ames is uniformly distributed between 1000
and 9000.
(a) What is the expected number of zombies in Ames?
(b) What is the variance in the number of zombies in Ames?
(c) What is the probability that between 3000 and 4000 zombies are in
Ames?
(d) What is the probability less than 6000 zombies are in Ames?
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(e) The zombie outbreak will cost the city $1,000,000 plus an additional
$17,000 per zombie. What is the expected cost of the zombie outbreak? What is the standard deviation of the cost of the zombie
outbreak?
(f) In the file http://gzt.public.iastate.edu/stat401/data/unifzombie.
txt, I have simulated the draws from this distribution. In JMP, load
this data set and calculate the mean and variance. Plot a histogram.
2. (only if we’ve gotten to the normal distribution)
The number of zombies in Ames is normally distributed with mean µ =
5000 and standard deviation σ = 2000.
(a) Write out the formula for the pdf for the number of zombies in Ames.
(b) What is the probability fewer than 5000 zombies are in Ames?
(c) What is the probability fewer than 7000 zombies are in Ames?
(d) What is the probability fewer than 3000 zombies are in Ames?
(e) How would we calculate the probability between 3256 and 8821 zombies are in Ames? Set up the equations, we do not need to evaluate
them.
(f) The zombie outbreak will cost the city $1,000,000 plus an additional
$17,000 per zombie. What is the expected cost of the zombie outbreak? What is the standard deviation of the cost of the zombie
outbreak?
(g) In the file http://gzt.public.iastate.edu/stat401/data/normalzombie.
txt, I have simulated the draws from this distribution. In JMP, load
this data set and calculate the mean and variance. Plot a histogram.
Make a normal quantile plot. There is something wrong with this
data, what is it? Look at the histogram or a scatter plot to see.
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References
• Mathematical Modeling of an Outbreak of Zombie Infection
• STAT 401 Page
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