Problem Set 5
Math 430/AMCS 530
Solutions are to be placed in my mailbox in DRL 4W1 by 5pm on 29 October, 2010.
(10 pts)
(1)
On average, only one person in a thousand has a particular rare blood type.
(a) Find the probability that in a city of 10,000 peop
Recall that we dened fkj as the probability of a state sj ever entering a state sk in a Markov
chain. We then used it to dene persistent and transient states:
Denition 0.1. A state si with fii = 1 is called persistent and is called transient if fii < 1.
F
Consider a collection of random variables cfw_Xi for 1 i n, with the same sample
space : that is,
Xi : R0 .
We can consider measuring all of these for a given outcome, whereupon we obtain
a collection of results (x1 , , xn ). Clearly, the set of all poss
Problem Set 8
Math 430/AMCS 530
Solutions are to be placed in my mailbox in DRL 4W1 by 5pm on 3 December, 2010.
(10 pts)
(1) Let (S, P ) be a Markov chain, with random variables Xi representing the steps (that
is, P (Xi = sj |Xi1 = sk ) = pjk ). Show that
Problem Set 7
Math 430/AMCS 530
Solutions are to be placed in my mailbox in DRL 4W1 by 5pm on 19 November, 2010.
(5 pts)
(1) Consider a BTP with 48 trials and probability 3/4 of success. Using the normal
approximation aorded by the central limit theorem,
Problem Set 6
Math 430/AMCS 530
Solutions are to be placed in my mailbox in DRL 4W1 by 5pm on 5 November, 2010.
(10 pts)
(1)
Consider an n-step Bernoulli trials process. Show that if U is a random variable that
depends only on the rst k trials, and V is a
Problem Set 4
Math 430/AMCS 530
Solutions are to be placed in my mailbox in DRL 4W1 by 5pm on 22 October, 2010.
(5 pts)
(1)
A high-school has 3000 students. Consider subsets of size 100 chosen by the following
methods:
(a) The rst 100 students who enter t
Problem Set 3
Math 430/AMCS 530
Solutions are to be placed in my mailbox in DRL 4W1 by 5pm on 15 October, 2010.
(5 pts)
(1)
Let E, F be events with positive probability of occuring, but simultaneously occur
with probability zero. Show that these are not i
Problem Set 2
Math 430/AMCS 530
Solutions are to be placed in my mailbox in DRL 4W1 by 5pm on 1 October, 2010.
(10 pts)
(1)
()
We extend the denition of the binomial coecient n to arbitrary real numbers x as
r
()
x
x(x 1) (x r + 1)
=
.
r
r!
Show( )the fol
Problem Set 1
Math 430/AMCS 530
Solutions are to be placed in my mailbox in DRL 4W1 by 5pm on 24 September, 2010.
(10 pts)
(1) Let A, B, C be arbitrary sets. Verify the following equalities
(a) (A B ) A B = (A B c ) (Ac B )
(b) (A B ) C = (A C ) (B C )
(c