Sabanc University
MATH 203, Recitation 11, Spring 2016-17
1. A continuous random variable X is said to have the normal distribution with parameters and > 0 if it has the
probability density
f (z) =
1
2
2
e(x) /(2 )
2
for < x < .
Show that
(a) f is symme
Sabanc University
MATH 203, Recitation 8, Spring 2016-17
1. Exercise 4.63: A game of chance is considered fair, or equitable, if each players expectation is equal to zero. If
someone pays us $10 each time that we roll a 3 or a 4 with a balanced die, how m
Sabanc University
MATH 203, Recitation 12, Spring 2016-17
1. A random variable X has a binomial distribution with n = 20 and = 0.4 Below you find a table of
numerical values for the probability distribution and cumulative distribution of a binomial distri
Sabanc University
MATH 203, Recitation 6, Spring 2016-17
1. Let X and Y be two discrete random variables with the joint probability distribution
f (x, y) =
1
(x + y),
21
for x = 1, 2, 3; y = 1, 2.
Find
(a) the marginal distribution of X;
(b) the condition
Sabanc University
MATH 203, Recitation 10, Spring 2016-17
1. If a contractors profit on a construction job can be looked upon as a continuous random variable having the
probability density
1
for 1 < x < 5
18 (x + 1)
f (x) =
0
elsewhere
where the units ar
Sabanc University
MATH 203, Recitation 7, Spring 2016-17
1. Suppose you would like to see a movie after school. In the movie theater that you would like to go to, there are
four movies scheduled to start at 9:00 p.m., 9:30 p.m., 10:00 p.m. and 10:30 p.m.
Sabanc University
MATH 203, Recitation 9, Spring 2016-17
1. Suppose you take a taxi to go to Viaport from the campus. Because of the traffic, the duration T of your trip is
random. It is a continuous random variable with the probability density
1
if 10 <
1
2
3
4
5
6
TOTAL
Name, Last Name:
Student Number:
Section or Assistant:
(SAMPLE)(SAMPLE) Math 204 Midterm II (SAMPLE)(SAMPLE)
April 2017
Time allowed is 1 hour 50 minutes. There are 6 problems worth a total of 30 points.
Justify your answers!
Problem 1:
DISCRETE MATHEMATICS - HOMEWORK WEEK 10
Relations
(1) Let R be an equivalence relation on the set A. In the lectures we saw that (i)(a, b)
R, (ii)[a] = [b], and (iii)[a][b] 6= are equivalent (Theorem 1), and we proved that
(i) (ii). Prove the remaining p
DISCRETE MATHEMATICS - HOMEWORK WEEK 12
Graphs 2
(1) The complementary graph G of a graph G = (V, E) is the graph with vertex set
V , where two vertices are adjacent in G if an only if there are not adjacent in G.
Describe the complements of the following
DISCRETE MATHEMATICS - HOMEWORK WEEK 8
Counting
(1) How many strings of 7 Turkish (lower case) characters are there?
(2) How many strings of 7 Turkish (lower case) characters start with a t, have an
r in the third position and do NOT end with an s?
(3) Ho
DISCRETE MATHEMATICS - HOMEWORK WEEK 12
Graphs 2
(1) Show that the concept of isomorphism between graphs defines an equivalence
relation on the set of simple graphs. An equivalence class of a graph with respect
to this equivalence relation is called an is
DISCRETE MATHEMATICS - HOMEWORK WEEK 9
Recurrence relations
(1) (a) Find a recurrence relation for the number an of words without aa using the
alfabet cfw_a, b, c, d.
(b) Find the characteristic roots of this recurrence relation.
(2) Consider a recurrence
DISCRETE MATHEMATICS - HOMEWORK WEEK 7
Induction and recursion
(1) Prove the following propositions by induction:
(a) P (n) :
n
X
i3 = (n(n + 1)/2)2 (n 1)
i=1
n
X
(b) P (n) :
(2i + 1)2 = (n + 1)(2n + 1)(2n + 3)/3 (n 0)
i=0
(c) P (n) :
n
X
i(i + 1) = n(n +
DISCRETE MATHEMATICS - HOMEWORK WEEK 11
Graphs 1
(1) Show that the sum, over the set of people at a party, of the number of people a
person has shaken hands with, is even. Assume that no one shakes his or her own
hand.
(2) Determine the number of edges in
Home Work#2 Math 204, Spring 2017
Feb 15:
7th Edition
Page 204: 53, 55, 56(greedy algorithm)
Pages 216-217: 1-9, 17, 18, 25, 27, 30, 39, topics regarding the question 30 will be
covered on Tuesday morning class
Pages 229-231: 8, 36(complexity of algorithm
Home Work Math 204, Spring 2017
Feb 8:
7th Edition(Global Edition)
Reading assignment: Example 5 on page 200(insertion sort ).
Pages: 204-205,
All exercises 3-24, 29(bubble sort), 33(insertion sort).
6th Edition
Reading assignment: Example 5 on page 174(i
DISCRETE MATHEMATICS - HOMEWORK WEEK 5
(1) Use the extended Euclidean algorithm to express the gcd(26, 91) as a linear combination of 26 and 91.
(2) Prove that the product of any three consecutive integers is divisible by 6.
(3) Determine an inverse of 7
Sabanc University
MATH 203, Recitation 5, Spring 2016-17
1. Exercise 3.96. In a certain city the daily consumption of water (in millions of liters) is a random variable
whose probability density is given by
1 xe x3
for x > 0
f (x) = 9
0
elsewhere
What ar
Sabanc University
MATH 203, Recitation 4, Spring 2016-17
1. Exercise 3.12: Find the distribution function of the random variable that has the probability distribution
f (x) =
x
,
15
2. Exercise 3.13: If X has the distribution function
0
1
4
1
F (x) =
2
3
Quiz 3
Math 204, Discrete Mathematics, Spring 2016-2017
Date: March 13, 2017, Time allowed is 20 minutes
Section:A
Problem
Problem
Problem
ac bd (
1. (2 points) Find prime factorization of 168.
2.(4 points) Let n be an integer. Show that gcd(n, n + 1) = 1
Quiz 3
Math 204, Discrete Mathematics, Spring 2016-2017
Date: March 13, 2017, Time allowed: 20 minutes
Section:B
Problem 1. (2 points) Find prime factorization of 252.
Problem 2.(4 points) Show that if a prime p divides an integer n, then p cannot divide
Math 203, Chapter 5
Hans Frenk (section A), Semih Sezer (section B)
Fall 2016
Chapter 5: Special Probability Distributions
I
Discrete Uniform Distribution
I
Bernoulli Distribution
I
Binomial Distribution
I
Negative Binomial and Geometric Distribution
I
Hy
Sabanc University
MATH 203, Recitation 2, Spring 2016-17
1. Exercise 2.41: A coin is tossed once. Then, if it comes up heads, a die is thrown once; if the coin comes
up tails, it is tossed twice more. Using the notation in which (H, 2), for example denote
Sabanc University
MATH 203, Recitation 3, Spring 2016-17
1. In a data network, there are two alternative paths between two nodes A and B. There are attackers on
both paths, and they randomly interfere and drop/steal files during transmission. Over each pa
Sabanc University
MATH 203, Recitation 1, Spring 2016-17
1. Exercise 1.1: An operation consists of two steps, of which the first can be made in n1 ways. If the first
step is made in the ith way, the second step can be made in n2i ways.
(a) Use a tree diag
1
2
3
4
BONUS
MATH 204
NAME
MIDTERM I
STUDENT NUMBER
TOTAL
MARCH 12, 2015
Time allowed is 100 minutes, Total points:50
You must show your work, give explanations and write clear solutions.
1 - (2+3+8 points)
a) Give a big-O estimate for each of the follow