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

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

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)

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

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

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

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]

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

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.