hw7 Probabilidade Harvard

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Stat 110 Homework 7, Fall 2015
Due: Friday 11/6 at 12:01 pm. Handwritten homework must be turned in on
paper. If you hand-write your homework, please turn it in at the Stat 110 dropbox
(outside SC 300K). If you typeset your homework, you can submit it online as a PDF or
on paper at the Stat 110 dropbox.
General instructions: The following problems are from Chapter 7 of the book. Unless
otherwise specified, show your work, simplify fully, and give clear, careful, convincing
justifications (using words and sentences to explain your logic, not just formulas). Be
sure to define your notation precisely. See the syllabus for the collaboration policy.
1. (BH 7.2) Alice, Bob, and Carl arrange to meet for lunch on a certain day. They arrive
independently at uniformly distributed times between 1 pm and 1:30 pm on that day.
Hint in advance: No calculus is needed for this problem. Keep in mind that for a Uni-
form in one dimension, probability is proportional to length, while for a Uniform in two
dimensions, probability is proportional to area.
(a) What is the probability that Carl arrives first?
For the rest of this problem, assume that Carl arrives first at 1:10 pm, and condition on
this fact.
(b) What is the probability that Carl will have to wait more than 10 minutes for one of
the others to show up? (So consider Carl’s waiting time until at least one of the others
has arrived.)
(c) What is the probability that Carl will have to wait more than 10 minutes for both
of the others to show up? (So consider Carl’s waiting time until both of the others have
arrived.)
(d) What is the probability that the person who arrives second will have to wait more
than 5 minutes for the third person to show up?
2. (BH 7.7) A stick of length L (a positive constant) is broken at a uniformly random
point X. Given that X = x, another breakpoint Y is chosen uniformly on the interval
[0, x].
(a) Find the joint PDF of X and Y . Be sure to specify the support.
(b) We already know that the marginal distribution of X is Unif(0, L). Check that
marginalizing out Y from the joint PDF agrees that this is the marginal distribution of
X.
(c) We already know that the conditional distribution of Y given X = x is Unif(0, x).
Check that using the definition of conditional PDFs (in terms of joint and marginal PDFs)
agrees that this is the conditional distribution of Y given X = x.
(d) Find the marginal PDF of Y .
(e) Find the conditional PDF of X given Y = y.
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3. (BH 7.23) The volume of a region in n-dimensional Euclidean space Rn is the integral
of 1 over that region. The unit ball in Rn is {(x1, . . . , xn) : x21 + · · ·+ x2n ≤ 1}, the ball of
radius 1 centered at 0. As mentioned in Section A.7 of the math appendix, the volume
of the unit ball in n dimensions is
vn =
pin/2
Γ(n/2 + 1)
,
where Γ is the gamma function, a very famous function which is defined by
Γ(a) =
∫ ∞
0
xae−x
dx
x
for all a > 0, and which will play an important role in the next chapter. A few useful facts
about the gamma function (which you can assume) are that Γ(a + 1) = aΓ(a) for any
a > 0, and that Γ(1) = 1 and Γ(1
2
) =
√
pi. Using these facts, it follows that Γ(n) = (n−1)!
for n a positive integer, and we can also find Γ(n + 1
2
) when n is a nonnegative integer.
For practice, please verify that v2 = pi (the area of the unit disk in 2 dimensions) and
v3 =
4
3
pi (the volume of the unit ball in 3 dimensions).
Let U1, U2, . . . , Un ∼ Unif(−1, 1) be i.i.d.
(a) Find the probability that (U1, U2, . . . , Un) is in the unit ball in Rn.
(b) Evaluate the result from (a) numerically for n = 1, 2, . . . , 10, and plot the results
(using a computer unless you are extremely good at making hand-drawn graphs). The
facts above about the gamma function are sufficient so that you can do this without doing
any integrals, but you can also use the command gamma in R to compute the gamma
function.
(c) Let c be a constant with 0 < c < 1, and let Xn count how many of the Uj satisfy
|Uj| > c. What is the distribution of Xn?
(d) For c = 1/
√
2, use the result of Part (c) to give a simple, short derivation of what
happens to the probability from (a) as n→∞.
4. (BH 7.25) Two companies, Company 1 and Company 2, have just been founded. Stock
market crashes occur according to a Poisson process with rate λ0. Such a crash would
put both companies out of business. For j ∈ {1, 2}, there may be an adverse event “of
type j,” which puts Company j out of business (if it is not already out of business) but
does not affect the other company; such events occur according to a Poisson process with
rate λj.
If there has not been a stock market crash or an adverse event of type j, then company
j remains in business. The three Poisson processes are independent of each other. Let X1
and X2 be how long Company 1 and Company 2 stay in business, respectively.
(a) Find the marginal distributions of X1 and X2.
(b) Find P (X1 > x1, X2 > x2), and use this to find the joint CDF of X1 and X2.
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5. (BH 7.29) Let X and Y be i.i.d. Geom(p), L = min(X, Y ), and M = max(X, Y ).
(a) Find the joint PMF of L and M . Are they independent?
(b) Find the marginal distribution of L in two ways: using the joint PMF, and using a
story.
(c) Find EM .
Hint: A quick way is to use (b) and the fact that L+M = X + Y .
(d) Find the joint PMF of L and M − L. Are they independent?
6. (BH 7.46) Each of n ≥ 2 people puts his or her name on a slip of paper (no two have
the same name). The slips of paper are shuffled in a hat, and then each person draws one
(uniformly at random at each stage, without replacement). Find the standard deviation
of the number of people who draw their own names.
7. (BH 7.56) You are playing an exciting game of Battleship. Your opponent secretly
positions ships on a 10 by 10 grid and you try to guess where the ships are. Each of your
guesses is a hit if there is a ship there and a miss otherwise.
The game has just started and your opponent has 3 ships: a battleship (length 4), a
submarine (length 3), and a destroyer (length 2). (Usually there are 5 ships to start, but
to simplify the calculations we are considering 3 here.) You are playing a variation in
which you unleash a salvo, making 5 simultaneous guesses. Assume that your 5 guesses
are a simple random sample drawn from the 100 grid positions.
Find the mean and variance of the number of distinct ships you will hit in your salvo.
(Give exact answers in terms of binomial coefficients or factorials, and also numerical
values computed using a computer.)
Hint: First work in terms of the number of ships missed, expressing this as a sum of
indicator r.v.s. Then use the fundamental bridge and naive definition of probability, which
can be applied since all sets of 5 grid positions are equally likely.
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