Chapter 5 - Distributions and Densities
5.1 Important Distributions
Results
Exercise 1
For which of the following random variables would it be appropriate to assign
a uniform distribution?
A Let X represent the roll of one die.
B Let X represent the number of heads obtained in three tosses of a coin.
C A roulette wheel has 38 possible outcomes: 0, 00, and 1 through 36. Let
X represent the outcome when a roulette wheel is spun.
D Let X represent the birthday of a randomly chosen person.
E Let X represent the number of tosses of a coin necessary to achieve a
head for the first time.
Answer
A, C and D are appropriate for a uniform distribution, since each outcome is equally likely.
This is not true for number of heads in three tosses, since e.g. getting 0 or 3 heads has a lower probability
than getting 1 or 2 heads. The number of tosses needed to get the first head also changes with number of tosses,
so it is not a uniform distribution.
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Exercise 2
Let n be a positive integer. Let S be the set of integers between 1 and n.
Consider the following process: We remove a number from S at random and
write it down. We repeat this until S is empty. The result is a permutation
of the integers from 1 to n. Let X denote this permutation. Is X uniformly
distributed?
Answer
Yes, provided that the number which is removed is selected completely randomly.
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Exercise 3
Let X be a random variable which can take on countably many values. Show
that X cannot be uniformly distributed.
Answer
If we assume that there is some p > 0 for each outcome, then when we sum them up
the probability becomes infinite which breaks the assumption that the sum of all outcomes
is 1 in a probability distribution. (Not showing the formal argument, but that is the gist of it).
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Exercise 4
Suppose we are attending a college which has 3000 students. We wish to
choose a subset of size 100 from the student body. Let X represent the subset,
chosen using the following possible strategies. For which strategies would it
be appropriate to assign the uniform distribution to X? If it is appropriate,
what probability should we assign to each outcome?
(a) Take the first 100 students who enter the cafeteria to eat lunch.
(b) Ask the Registrar to sort the students by their Social Security number,
and then take the first 100 in the resulting list.
(c) Ask the Registrar for a set of cards, with each card containing the name
of exactly one student, and with each student appearing on exactly one
card. Throw the cards out of a third-story window, then walk outside
and pick up the first 100 cards that you find.
Answer
(a)
This will not be a proper uniform distribution. People who are sick, away, that never eat in the cafeteria etc.
will not be in the pool.
(b)
SSN has some geographical and other logic ingrained in the number, so it is not completely random.
(c)
This seems appropriate for a uniform distribution. Each of the 100 cards will have a 1/3000 probability
at the beginning.
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Exercise 5
Under the same conditions as in the preceding exercise, can you describe
a procedure which, if used, would produce each possible outcome with the
same probability? Can you describe such a procedure that does not rely on a
computer or a calculator?
Answer
Not going into simulation details, but in R:
sample(1:3000, size=100, replace=FALSE)
For a method not requiring a computer, we make numbered cards of all students from 0 to 2999.
We use a 3-sided and a 10-sided die for the randomness. Which are tossed
1 times and 3 times. The values 3, 10 become 0. So if we get: 2, 7, 2, 5, this points to 2725.
Alternatively, we could put all the numbered cards in an urn and draw 100 at random.
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Exercise 6
Let X
1, X
2, ..., X
n be n mutually independent random variables, each of
which is uniformly distributed on the integers from 1 to k. Let Y denote the
minimum of the X
i’s. Find the distribution of Y .
Answer
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Exercise 7
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