hw6 Probabilidade

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Stat 110 Homework 6, Fall 2015
Due: Friday 10/30 (Halloween Eve) at 12:01 pm. Handwritten homework must
be turned in on paper (there were too many scanned homeworks that were hard to
read in Canvas). 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 Chapters 5–6 of the book
Introduction to Probability by Joe Blitzstein and Jessica Hwang. Unless otherwise
specified, show your work, simplify fully, and give clear, careful, convincing justifica-
tions (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 5.6) The 68-95-99.7% rule gives approximate probabilities of a Normal r.v. being
within 1, 2, and 3 standard deviations of its mean. Derive analogous rules (i.e., find
the appropriate replacements for the 68, 95, and 99.7) for the following distributions.
(a) Unif(0, 1).
(b) Expo(1).
(c) Expo(1/2). Discuss whether there is one such rule that applies to all Exponential
distributions, just as the 68-95-99.7% rule applies to all Normal distributions, not just
to the standard Normal.
2. (BH 5.39) Three students are working independently on their probability homework.
All 3 start at 1 pm on a certain day, and each takes an Exponential time with mean
6 hours to complete the homework. What is the earliest time when all 3 students will
have completed the homework, on average? (That is, at this time all 3 students need
to be done with the homework.)
Hint: This can be done without using calculus, by drawing a timeline and using
important properties of the Exponential.
3. (BH 5.42) (a) Fred visits Blotchville again. He finds that the city has installed an
electronic display at the bus stop, showing the time when the previous bus arrived.
The times between arrivals of buses are still independent Exponentials with mean 10
minutes. Fred waits for the next bus, and then records the time between that bus and
the previous bus. On average, what length of time between buses does he see?
(b) Fred then visits Blunderville, where the times between buses are also 10 minutes on
average, and independent. Yet to his dismay, he finds that on average he has to wait
more than 1 hour for the next bus when he arrives at the bus stop! How is it possible
that the average Fred-to-bus time is greater than the average bus-to-bus time even
though Fred arrives at some time between two bus arrivals? Explain this intuitively,
and construct a specific discrete distribution for the times between buses showing that
this is possible.
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4. (BH 5.61) Let X1, X2, . . . be the annual rainfalls in Boston (measured in inches) in
the years 2101, 2102, . . . , respectively. Assume that annual rainfalls are i.i.d. draws
from a continuous distribution. A rainfall value is a record high if it is greater than
those in all previous years (starting with 2101), and a record low if it is lower than
those in all previous years.
(a) In the 22nd century (the years 2101 through 2200, inclusive), find the expected
number of years that have either a record low or a record high rainfall.
(b) On average, in how many years in the 22nd century is there a record low followed
in the next year by a record high?
(c) By definition, the year 2101 is a record high (and record low). Let N be the number
of years required to get a new record high. Find P (N > n) for all positive integers n,
and use this to find the PMF of N .
5. (BH 6.15) Let W = X2 + Y 2, with X, Y i.i.d. N (0, 1). The MGF of X2 turns out
to be (1− 2t)−1/2 for t < 1/2 (you can assume this).
(a) Find the MGF of W .
(b) What famous distribution that we have studied so far does W follow (be sure to
state the parameters in addition to the name)? In fact, the distribution of W is also
a special case of two more famous distributions that we will study in later chapters!
6. (BH 6.24) Let X and Y be i.i.d. Expo(1), and L = X−Y . The Laplace distribution
has PDF
f(x) =
1
2
e−|x|
for all real x. Use MGFs to show that the distribution of L is Laplace.
7. (BH 6.25) Let Y = Xβ, with X ∼ Expo(1) and β > 0. The distribution of Y is
called the Weibull distribution with parameter β. This generalizes the Exponential,
allowing for non-constant hazard functions. Weibull distributions are widely used in
statistics, engineering, and survival analysis; there is even an 800-page book devoted
to this distribution: The Weibull Distribution: A Handbook by Horst Rinne.
For this problem, let β = 3.
(a) Find P (Y > s+ t|Y > s) for s, t > 0. Does Y have the memoryless property?
(b) Find the mean and variance of Y , and the nth moment E(Y n) for n = 1, 2, . . . .
(c) Determine whether or not the MGF of Y exists.
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