December 19, 2009
CS201 (Intro. to Computing) MIDTERM II
1
2
Name
ID
3
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5
6
7
TOTAL
: Solutions
:
Notes: a) Please answer the questions only in the provided space after each question.
b) Duration is 100 minutes.
c) Closedbook, closednotes, no calculato
Exam I, Spring 201617
March 10, 2017, 19:40  21:40
Name:
Student Number:
READ ALL THE ITEMS BELOW BEFORE YOUR START THE
EXAM
All unauthorized materials (textbooks, notes, electronic devices, bags, etc.), and in
particular mobile phones and smart watche
Exam II, Spring 201617
April 14, 2017, 19:40  21:40
Name:
Student Number:
READ ALL THE ITEMS BELOW CAREFULLY BEFORE YOUR
START THE EXAM
All unauthorized materials (textbooks, notes, electronic devices, bags, etc.), and in
particular mobile phones and s
Exam III, Spring 201617
May 10, 2017, 19:40  21:40
Name:
Student Number:
READ ALL THE ITEMS BELOW CAREFULLY BEFORE YOUR
START THE EXAM
All unauthorized materials (textbooks, notes, electronic devices, bags, etc.), and in
particular mobile phones, table
RECITATION 1 PROBLEMS
SUMATH 203
SUMMER 2014/15
Problem 1. Two dice are thrown. Let E be the event that the sum of the dice is odd; let F be the event
that at least one of the dice lands on 1; and let G be the event that the sum is 5. Describe the events
RECITATION 2 PROBLEMS
SUMATH 203
SUMMER 2014/15
Problem 1. (a) How many dierent 7place license plates are possible if the rst 2 places are for letters
and the other 5 for numbers?
(b) Repeat part (a) under the assumption that no letter or number can be
SUMATH 203
RECITATION 3 PROBLEMS
SUMMER 2014/15
Problem 1. Suppose you pick a card at random from a well shued deck. What is the probability that
it is an Ace given that it is a Diamond?
Problem 2. Three cards are drawn in succession, without replacement
RECITATION 4 PROBLEMS
SUMATH 203
SUMMER 2014/15
Problem 1. Toss a coin 3 times and let X be the number of Heads observed. Find
(a) The range of X (b) The p.d.f of X (c) P(X = 2) (d) P(X < 2).
Problem 2. For each of the following, determine whether the gi
Ch. 4, Section 1: Mathematical Expectation E(X)
A lot of 12 television sets includes 2 with white cords. If 3 of the sets are chosen at random for
shipment to a hotel, how many sets with white cords can the shipper expect to send to the hotel?
Answer: 1/2
Ch. 3, Section 1: Random Variables
In Ch. 2 we considered probability spaces I A sample space , I A set of events F, consisting of subsets
of I A probability measure assigned to each event.
Denition (Random Variable) A random variable is a function X : S,
Ch. 2: Section 1 and 2: Introduction and Sample Spaces.
In the rst subsection the concept of a random experiment by means of examples is introduced. The
exposition is somewhat more elaborate than that in the textbook. Denition (Random experiment) A
random
SUMATH 203
RECITATION 5 PROBLEMS
SUMMER 2014/15
Problem 1. Let X be a discrete random variable. We know that E(X) = 1/4 and E(X 2 ) = 1/2.
(a) Find E(2X + 1) and E([X + 1]2 ).
(b) Let f (x) be the probability distribution function for X. Suppose f (x) >
MATH203  Homework 6
Muzaffer Akat
You do not have to turn in.
1. (4.5) Given two random variables X and Y , and use Theorem 4.4 to express E(X) in
terms of
(a) the joint density of X and Y ;
(b) the marginal density of X.
2. (4.6) Find the expected value
MATH203  Homework 8  Solutions
Muzaffer Akat
Some useful information:
k
P
u = k(k+1)
2
u=1
k
P
u=1
P
x=0
u2 =
k(k+1)(2k+1)
6
ux =
1
,
1u
0<u<1
1. (5.1)
1
, x = 1, 2, ., k
k
k
X
1
1 k(k + 1)
k+1
1
=
=
x = (1 + 2 + . + k) =
k
k
k
2
2
x=1
f (x) =
2
2
k
X
Sabanci University
MATH 203, Recitation 4, Fall 2015
1. Exercise 3.2: For each of the following, determine whether the given function can serve as the probability
distribution of a random variable with the given range.
(a) f (x) =
(b) f (x) =
(c) f (x) =
MATH203  Homework 9
Muzaffer Akat
You do not have to turn in.
1. (5.63) A quality control engineer inspects a random sample of two handheld calculators from each incoming lot of size 18 and accepts the lot if they are both in good
working condition; oth
MATH203  Homework 8
Muzaffer Akat
You do not have to turn in.
1. (5.1) If X has the uniform distribution f (x) =
(a) its mean is =
1
k
for x = 1, 2, . . . , k show that
k+1
;
2
(b) its variance is 2 =
k 2 1
.
12
2. (5.2)If X has the discrete uniform dist
MATH203  Homework 11
Muzaffer Akat
You do not have to turn in.
1. (6.31) Show that the normal distribution has
(a) a relative maximum at x = ;
(b) the inflection points at x = and x = + .
2. (6.33) Using the moment generating function of the normal distr
MATH203  Homework 10
Muzaffer Akat
You do not have to turn in.
1. (6.4) Show that if a random variable has a uniform density with the parameters and
, the r th moment about the mean equals
(a) 0 when r is odd;
r
1
(b) 1+r
when r is even.
2
2. Let X be a
MATH203  Homework 10  Solutions
Muzaffer Akat
You do not have to turn in.
R
1
1
]r dx = ()2
[x +
[2x ( + )]r dx
1. (6.4) r =
r
2
[2x(+)]r+1
1

= ()2
r
2(r+1)
R
=
()r+1 (1)r+1 ()r+1
1
()2r
2(r+1)
(a) = 0 when r is odd;
r
1
(b) = 1+r
when r is even.
MATH203  Homework 11  Solutions
Muzaffer Akat
1. (6.31) We need to find the relative maximum, and inflection points of the function
f (x) =
1 x 2
1
e 2 ( )
2
Also define the function
1 x 2
1 x 2
) =C (
)
g(x) = log(f (x) = log( 2) (
2
2
where C = log
MATH203  Homework 7
Muzaffer Akat
You do not have to turn in.
2
1. (4.40) Given the moment generating function MX (t) = e3t+8t , find the moment generating function of the random variable Z = 14 (X 3), and use it to determine the
mean and the variance of
Sabanci University
Faculty of Engineering and Natural Sciences
Mathematics
MATH 203, Final Exam
May 23, 2015, 9:30  11:30
Name:
Student Number:
Q1
Q2a
Q2b
Q2c
Q2d
Q3a
Q3b
Q3c
Q4
Q5a
Q5b
Q5c
Total
2
2
3
3
3
3
2
2
3
2
2
3
Max = 30
There are five questions