Exam 1
Math 430/AMCS 530
Instructions
(1) Write your name in the space provided below.
(2) This is a closed-book exam.
(3) No calculators or cheat-sheets are allowed. Be sure to show all of your work in each
problem.
Name:
1
Possible
2
3
4
5
6
5
15
10
10
1) Give an example of a random variable that has
(a) The binomial distribution b(n, p, k ) what are n, p, and k ?
(b) The uniform distribution
(c) The poisson distribution p(k ; ) what are k and ?
(d) The hypergeometric distribution
2) N sticks of length
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
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 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 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 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
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 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 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