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Problem Set 4

1. This problem introduces the "drunken sailor problem" and random walks. It asks the student to calculate: a) The expected value of the sailor's position relative to the pub after N steps b) The variance of the sailor's position c) Whether the standard deviation is the same as the average distance between the sailor and pub. 2. This problem repeats the drunken sailor problem but with a probability p of stepping right and 1-p of stepping left. 3. This problem asks the student to find the value of λ that maximizes the probability P{X=k} for a Poisson random variable X with parameter λ and k>0.

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48 views2 pages

Problem Set 4

1. This problem introduces the "drunken sailor problem" and random walks. It asks the student to calculate: a) The expected value of the sailor's position relative to the pub after N steps b) The variance of the sailor's position c) Whether the standard deviation is the same as the average distance between the sailor and pub. 2. This problem repeats the drunken sailor problem but with a probability p of stepping right and 1-p of stepping left. 3. This problem asks the student to find the value of λ that maximizes the probability P{X=k} for a Poisson random variable X with parameter λ and k>0.

Uploaded by

Qinxin Xie
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Problems for Session 4, “Special Probability Distributions”

1. “Drunken Sailor Problem” (This is also an introduction to “random walks” or


“Brownian motion,” which is also the essence of stochastic differential equations.) A
sailor gets out of the pub and walks along the street. Each of his steps is 1 meter long,
and he makes them randomly to the left or to the right, with probability 0.5. After N
steps, he falls asleep. Let X be a random variable, corresponding to his position relative
to the pub after N steps. (So X=-5 corresponds to being 5 meters to the left from the pub,
X=3 corresponds to being 3 meters right from the pub, and X=0 corresponds to coming
back to the pub. Thus X is a discrete random variable and its possible values are
integers from –N to N; these possible values are even if N is even, and they are odd if N
is odd.) Calculate:
a) expected value of X (i.e., E[X]),
b) variance of X, (i.e., V(X)),
c) standard deviation of X. Is it the same as the “average distance between the
sailor and the pub after N steps?”

2. Repeat problem 1, but now assume that each step is done with probability p to the right
and probability (1-p) to the left.

3. (Ross, Chapter 4, Theoretical Exercise 18, p.181.) Let X be a Poisson random variable
with parameter . What value of  maximizes P{X=k}, k>0?

4. (Ross, Chapter 4, Problem 40, p. 176.) On a multiple-choice exam with 3 possible


answers for each of the 5 questions, what is the probability that a student would get 4 or
more correct answers just by guessing?

5. (Ross, Chapter 4, Problem 41, p. 176.) A man claims to have extrasensory perception. As
a test, a fair coin is flipped 10 times, and he is asked to predict the outcome in advance.
Our man gets 7 out of 10 correct. What is the probability that he would have done at
least this well if he had no extrasensory perception?

6. Suppose that the probability density function for the market share s of a new product in
a relatively new market is as follows, where s is expressed as a proportion of the
market:
f(s)=ks(1-s)2 for 0<s<1, f(s)=0 elsewhere.
a) Find k.
b) What is the probability that the market share is less than one-half?
c) If the company’s market share goes down by 30% to s*=0.7s, find the expected
value of s*.
d) Find the standard deviations of s and s*.

7. (Ross, Chapter 5, Problem 13, p. 229, slightly modified.) You arrive at a bus stop at 10
o’clock, knowing that the bus will arrive at some time uniformly distributed between 10
and 10:30.
a) What is the probability that you will have to wait longer than 10 minutes? How
much you will have to wait on average?
b) If at 10:15 the bus has not yet arrived, what is the probability that you will have
to wait at least 10 minutes more?
c) Generalize the last two questions: for any x>0, y>0 find the probability that 1) you
will not have to wait more than x minutes; 2) if the bus has not come after you
waited for x minutes, what is the probability that you have to wait no more than
y additional minutes?
Comment: in the last case, be careful with x>30 and/or x+y>30. Make sure your
answers/formulas/graphs agree with your intuition.

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