December 19, 2009
CS201 (Intro. to Computing) MIDTERM II
1
2
Name
ID
3
4
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
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
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 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
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
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
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