Summer 2015
Franklin
ENEE 324
MIDTERM EXAM
7 July 2015
Note: 1 hour, 20 minutes duration; Closed book, Switched off cell phones
1. (4 points) An experiment consists of picking 3 cards at random from a shuffled deck of 52
cards.
(a) How many outcomes are i

LECTURE 1 - COMPREHENSION
BASIC TERMINOLOGY
1. If the random page consisted of 30 lines with 60 characters each,
and if we allowed additional characters A-Z (capitals) and
three punctuation marks . , ? , what would the size of the sample
space be?
2. (

Leave blank
Grade:
LAST NAME:
First Name:
ENEE 324 SPRING 2012
SECOND EXAMINATION
PROBLEM 1 (15 pts.)
Grade:
The pdf of a random variable X is given by
fX (x) =
1
1
c
(x) + (x 2) + 3 u(x 1)
4
4
x
fX(x)
1/4
1/4
0
1
2
x
(i) (4 pts.) Evaluate c.
(ii) (7 pts

Leave blank
Grade:
LAST NAME:
First Name:
ENEE 324 FALL 2011
SECOND EXAMINATION
PROBLEM 1 (15 pts.)
Grade:
The pdf of a random variable X is shown below.
fX(x)
c
c
0
1
2
3
x
(i) (3 pts.) Evaluate c.
(ii) (6 pts.) Make a detailed sketch of the cdf FX (x).

_
Leave blank
_ _ _ _
Grade:
*
LAST NAME:
First Name:
ENEE 324 FALL 2011
SECOND EXAMINATION
PROBLEM 1 (15 pts.)
The pdf of a random variable X is shown below.
Grade:
fx)
C
.0
1C
1
23
x
(i) (3 pts.) Evaluate c.
(ii) (6 pts.) Make a detailed sketch of the c

Alternative Problem 13
The probability density function (pdf) of a continuous random variable X is given by
cx,
x [0, 1]
c,
x (1, 2)
fX (x) =
c(3
x),
x
[2, 3]
0,
elsewhere
(i) Sketch fX (x). What is the range of values of X?
(ii) What is the value of c?

PROBLEM 1 (15 pts.)
The pdf of a random variable X is shown below.
lit-"3
"
0 I 2 3 .r
(i) (5 pts.) If EIX] m 2, evaluate the constants u and I). (Use these values in what follows.)
(ii) (6 pts.) Determine and sketch the cdf Fx(:t:).
(iii) (4 pts.) A fai

ENEE 324 FALL 2011
FIRST EXAMINATION
PROBLEM I (15 pts.) ' Grade:
Events A1, A2 and A3 are such that:
P[A1-] = 0! (every 2')
P[Ai I Aj] *" (3 (every distinct i, j)
P[Az' I Aj 0 Are] = 2/ (every distinct i, j and k)
(i) (3 pts.) Determine P[A1 (1 A2 n A3

CL
PROBLEM5
J
I
(,i
I
L
0.
4
(
4
I)
ii
iT\
L
1
I
4
(1
Ci
_
II
3
L
Li
A
(Xc
sI
II
4c
Ii
4
;1
p.,
t9
0
4.
IL
I)
A
Cv)
0
sI
9
01
4.
il
1
4
e4
1)
0
(V
4
0
Ii
A
QI
j
4
I)
$w
$
(1
I
S
U
C
$
0
Li
L
L
1
/
a,
ci
4
I)
[1

PROBLEM9
ii
$
4
oI
4 S.
I
Li
V
4
0
(4
0
IN.
0
N
h
0
o
C,)
(4
4
Qc3
cJ
2
ii
Li
cto,
I,
CbO
Iv)
I
11
S.
5
44
r.J
A
I
_%
4.
Cl
_%
I
cJ
0
R
N3
C,
\ls
0
J
zb
c
0
v
C
0
N
0
\
c\i
(
\
%
(.j
%
oc
!3
(N
3
r
R%
,
N
pI
.
J
r
0
I
I
I
q3
0
0
if:
F
c1
L
I
II
%c4;
C
U

PROBLEM4
cJ
L)
J
e2
o
L
1
.c
0
(I
?
C
FN
I
\J
ii
lit:)
C)
II
0
Th
r)
.
U
r)
tJ
o
0
ii
C
C
cC
C)
.4
Li
e
Qc
1)
L4
.
Cc
i;:
C
C
c)
J
I
+
C
)
c
ccj
Li
C
I)
c
U
L
I
0
I
0
C
0
C
0
.
0
c:j
S
c
0_
11
C
1JL
C)
Ii
0
r7
1
c
c
1
oai
0
0
I
Li

Leave blank
Grade:
*
LASTNAME:
First Name:
ENEE 324 FALL 2011
FIRST EXAMINATION
Grade:
PROBLEM 1 (15 pts.)
Events A
, A
1
2 and A
3 are such that:
PIA?] =
A] =
1
P[A
P[AjIAJ flAk]
(every i)
(every distinct i,
(every distinct i,
=
1
A
j)
j
and k)
(i) (3 pt

Solved Example 1
Consider a random experiment in which a coin (marked H and T , as usual) is tossed six times.
The result of each toss is recorded in a sequence.
i) What is the sample space S for this experiment? What is the smallest set D such that S = D

Problem 1
A meteorite strikes Earth at a random point of latitude 1 and longitude 2 , both of which are
measured in degrees. The range of 1 is [90, 90], where positive values denote north. The range
of 2 is (180, 180], where positive values denote east.
(

Summer 2014
Franklin
ENEE 324
MIDTERM EXAM
1 July 2014
Note: 1 hour, 20 minutes duration; Closed book, Switched off cell phones; No calculators
1. (4 points) The children born in a country is equally likely to be a boy or a girl. A family has 2
children.

Discrete Random Variables
Reading: Chapter 2.1, 2.2, and 2.3
Random Variable
A random variable, X, consists of (1) an
experiment with a probability measure
defined on the sample space S and (2) a
function that maps each outcome s in S to a
real number X(s

Continuous Random Variables
Reading: Chapter 3.1 3.8
Homework: 3.1.2, 3.2.1, 3.2.4, 3.3.2, 3.3.7,
3.4.4, 3.4.5, 3.4.9, 3.5.3., 3.5.6,
3.6.1, 3.6.6, 3.7.1, 3.7.3, 3.7.11, 3.8.1.
Cumulative Distribution Function
CDF of a random variable X is: FX(x) = P[Xx]

Set Theory, Sample Space,
and Event Space
Reading: Chapter 1.1 and 1.2
Set and Element
A set is a collection of things, objects, or
elements.
Examples:
A=cfw_1,2,3,4,5,6
B=cfw_January, February, May, July
C=cfw_all months that end with a y
D=cfw_x|x is a

Continuous Random Variables
Reading: Chapter 3.1 3.8
Homework: 3.1.2, 3.2.1, 3.2.4, 3.3.2,
3.3.7,
3.4.4, 3.4.5, 3.4.9, 3.5.3., 3.5.6,
3.6.1, 3.6.6, 3.7.1, 3.7.3, 3.7.11, 3.8.1.
Cumulative Distribution Function
CDF of a random variable X is: FX(x) = P[Xx]

Pairs of Random Variables
Joint Cumulative Distri.
Function
For r.v.s X and Y, the joint CDF is
FX,Y(x,y) = P[Xx, Yy]
Properties of joint CDF
0 FX,Y(x,y) 1
FX,Y(x,) = FX(x), FX,Y(,y) = FY(y)
FX,Y(x,-) = FX,Y(-,y) =0
FX,Y(,)=1
FX,Y(x,y) FX,Y(x,y) if xx and

5. Random Vectors
Prob. Models of n R.V.s
Single Random Variable:
CDF: FX(x) = P[Xx]
2 Random
Variables:
PMF: PX(x) = P[X=x]
PDF: fX(x) = (d/dx) FX(x)
N Random Variables:
Joint CDF: FX1,Xn(x1,xn) =
Joint CDF:
FX,Y(x,y)
= P[Xx, Yy]
Joint PMF:
PX,Y(x,y)
= P

Set Theory, Sample Space,
and Event Space
Reading: Chapter 1.1 and 1.2
Set and Element
A set is a collection of things, objects, or
elements.
Examples:
A=cfw_1,2,3,4,5,6
B=cfw_January, February, May, July
C=cfw_all months that end with a y
D=cfw_x|x is a

6. Sums of Random Variables
PDF of W=X+Y
R.V.s X and Y have joint PDF fX,Y(x,y).
What is fW(w) for W=X+Y?
CDF: FW(w) = P[X+Yw] = dxw-x fX,Y(x,y)dy
PDF: fW(w) = (d/dw)FW(w) = fX,Y(x,w-x)dx
(d/dx) h2(x)
h1(x) f(t)dt = f(h2(x)h2(x)-f(h1(x)h1(x)
Th. 6.4 fW(w)

Discrete Random Variables
Reading: Chapter 2.1, 2.2, and 2.3
Random Variable
A random variable, X, consists of (1) an
experiment with a probability measure
defined on the sample space S and (2) a
function that maps each outcome s in S to a
real number X(s

Alternative Problem 1
Let i be the noon temperature in College Park on Day i (where i = 1, 2, . . .), taking values between
20 F and 100 F and recorded with infinite precision.
(i) Suppose that the noon temperatures on Days 1 and 2 are available. On the (

ENEE 324 FIRST EXAMINATION: PRACTICE PROBLEMS
PROBLEM 1
A jar contains five cookies. Alice wakes up in the middle of the night and takes i cookies out of
the jar, where i can be any number from 0 to 5 (inclusive) with equal probability (1/6). Later on,
Bo

Leave blank
Grade:
LAST NAME:
First Name:
ENEE 324 FALL 2011
FIRST EXAMINATION
PROBLEM 1 (15 pts.)
Grade:
Events A1 , A2 and A3 are such that:
P [Ai ] =
P [Ai | Aj ] =
P [Ai | Aj Ak ] =
(every i)
(every distinct i, j)
(every distinct i, j and k)
A1
A2

Leave blank
Grade:
LAST NAME:
First Name:
ENEE 324 SPRING 2012
FIRST EXAMINATION
PROBLEM 1 (15 pts.)
Grade:
Consider a random point on the square [0, 5]2 whose position has a uniform probability assignment
over that square. The coordinates of that point

Answer: Clearly, the word length k must be less than or equal to
n. Reasoning as before, we have
Recitation 0
Basic Counting Methods
n 1-letter words
We have a set A of n distinct elements, also called alphabet :
n(n 1) 2-letter words
A = cfw_1, 2, . . .