4. Random Vectors
Often a single experiment will have more than one random variable which is of interest.
Since they result from the same experiment, these random variables are usually grouped together as a random vector.
We shall consider the case whe
Practice MTH 511 Test #2
SLfoINJ_.i
Last Name Print): 1
Part A. l\/Iultip1e Choice marks each]
In the multiple choice questions below. only one alternative is correct in each case so
check all alternatives carefully before circling the correct one.
1.
a
Ryerson University
Lab 10 - MTH 514 - Fall 2014
1. Flip a coin twice. On each ip, the probability of heads equals p. Let Xi
equal the number of heads (either 0 or 1) on ip i. Let W = X1 X2 and
Y = X1 + X2 . Find pW,Y (w, y) and pW/Y (w/y).
Solution
Notice
Ryerson University
Lab 7 - MTH 514 - Fall 2014
1. The voltage across a 1 resistor is a uniform random variable on the
interval [0, 1]. The instantaneous power is Y = X 2 . Find the c.d.f. FY (x)
and the p.d.f. fY (x) of Y .
Solution Notice that 0 Y 1, the
RYERSON UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH 514Stochastic Processes
Midterm 1 version A
Last Name (Print):
Signature:
Date: Sept. 26, 2008
Duration: 1.5 hour
. First Name:
. Student Number:
.
Professor
M. Escobar-Anel
G. Ord
Instructions:
1. This is
Review of Lecture 11
3.6 Delta Functions, Mixed Random Variables
N. Ruzgar
MTH514 W14 Lecture 13, 14
1/25
MTH514 Probability and Stochastic Processes
Lecture 13, 14
3.7 Probability Models of Derived Random Variables
3.8 Conditioning a Continuous Random Va
Review of Lecture 7, 8
2.5 Averages
2.6 Functions of a Random Variables
2.7 Expected Value of a Derived Random Variable
2.8 Variance and Standard Deviation
N. Ruzgar
MTH514 W14 Lecture 9, 10
1/54
MTH514 Probability and Stochastic Processes
Lecture 9,
Review of Lecture 1, 2
1.1 Set Theory
1.2 Applying Set Theory to Probability
1.3 Probability Axioms
1.4 Some Consequences of the Axioms
N. Ruzgar
MTH514 W14 Lecture 3, 4
1/56
MTH514 Probability and Stochastic Processes
Lecture 3, 4
1.5 Conditional Probabi
MTH514 W14 Weekly Schedule
1
2
3
4
5
6
Date
Jan 6; Jan 8
Jan 13; Jan 15
Jan20; Jan 22
Jan 27; Jan 29
Feb 3; Feb 5
Feb10; Feb12
7
8
9
10
11
12
13
Feb24; Feb 26
March 3; March 5
March10; March 12
March17; March 19
March24; March 26
March 31; Apr 2
Apr 7; Ap
Ryerson University
Mathematics Department
MTH514 Probability and Stochastic Processes - W14
Dr. Nursel Ruzgar
E-mail: nruzgar@ryerson.ca
Office: VIC707
Office extension: 3173
Office Hours: Mondays: 13:15-14:45pm, VIC707
Textbook: Probability and Stochasti
Ryerson University
Lab 9 - MTH 514 - Fall 2014
1. The pair of random variables (X, Y ) has joint p.m.f.:
pX,Y
x = 1
x=1
y=0
1/4
0
y=1
1/8
1/2
y=2
0
1/8
a) Find the margin p.m.f. pX (x) and pY (y).
b) Find the expected values E[X] and E[Y ].
c) Find the st
Ryerson University
Lab 8 - MTH 514 - Fall 2014
1. Let Y be an exponential distribution with parameter = 0.2. Given the
event A = [Y < 2].
a) What is the conditional p.d.f., fX/A (x)?
b) Find the conditional expected value, E[X/A].
Solution a) We rst compu
Review of Lecture 13, 14
3.7 Probability Models of Derived Random Variables
3.8 Conditioning a Continuous Random Variable
4.1 Joint Cumulative Distribution Function
4.2 Joint Probability Mass Function
4.3 Marginal PMF
N. Ruzgar
MTH514 W14 Lecture 15,
Definition: joint PMF: PX,Y(x,y)=P[X=x,Y=y]
3.1For any subset B of X,Y plane,P of eventcfw_(X,Y
P B
( x ,y B )
P
X ,Y
x
,
PX ,Y PX( x )
P
y
X ,Y
( x, y )
Def. for any
PX ,Y( x, y )
3.4for discrete random variable W=g(X,Y)
x SX y SY
g( x, y )PX ,Y( x, y
RYERSON UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH 514Stochastic Processes
Midterm 2 version A
Last Name (Print):
Signature:
Date: Oct. 22, 2007
Duration: 1.5 hour
. First Name:
. Student Number:
.
Professor (circle one)
M. Escobar-Anel
G. Ord
Instructions:
.0er
RYERSON UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH 514 - Probability and Stochastic Processes
First Term Test - Winter 2011
Last Name (print): R50
First Name (print):
Student ID Number:
Signature:
Instructor:
Escobar
Date: February 3, 2011, 2:
Ryerson University
Lab 6 - MTH 514 - Fall 2014
1. Suppose that X is a continuous random variable with density function
3x5
otherwise
Kx
0
fX (x) =
a) Find the value of constant K.
b) Find the probability P (X < 4).
c) Find the probability that X is smalle
Ryerson University
Lab 1 - MTH 514 - Fall 2014
1. A manufacturer of front lights for automobiles tests lamps under a high
humidity, high temperature environment using intensity and useful life as
the responses of interest. The following table shows the pe
Ryerson University
Lab 2 - MTH 514 - Fall 2014
1. Mobile telehpones perform handos as they move from cell to cell. During
a call, either performs zero handos(H0 ), one hando (H1 ), or more than
one hando( H2 ). In addition, each call is either long (L) or
Ryerson University
Lab 3 - MTH 514 - Fall 2014
1. A discrete random variable X has probability mass function:
p(x) =
C(2x + 1)
0
x = 0, 1, 2, 3, 4
otherwise
a) Compute the value C.
b) Compute P (X = 4).
c) Find the probability that the random variable tak
Ryerson University
Lab 5 - MTH 514 - Fall 2014
1. It can take up to four days after you call for service to get your computed
repaired. The computer company charges for repairs according to how
long you have to wait. The number of days D until the service
Ryerson University
Lab 12 - MTH 514 - Fall 2014
1. Queries presented to a computer database are a Poisson process of rate
= 6 queries/min. An experiment consists in monitoring the database for
m minutes and recording Nm , the number of queries presented.
Ryerson University
Lab 11 - MTH 514 - Fall 2014
1. Random sample distribution
Let (Xn ) be a sequence of independent random variables with the same
probability distribution, E(Xn ) = and V ar(Xn ) = 2 and dene the
sample average as:
n
1
Xk
Xn =
n
k=1
i) F
Review of Lecture 21
6.4 MGF of the Sum of Independent Random Variables
6.6 Central Limit Theorem
N. Ruzgar
MTH514 W14 Lecture 23, 24
1/21
MTH514 Probability and Stochastic Processes
Lecture 23, 24
6.7 Applications of Central Limit Theorem
10.2 Types of S
Review of Lecture 9, 10
3.1 The Cumulative Distribution Function
3.2 Probability Density Function
3.3 Expected Values
3.4 Families of Continuous Random Variables
3.5 Gaussian (Normal) Random Variables
N. Ruzgar
MTH514 W14 Lecture 11
1/9
MTH514 Probability
Review of Lecture 5, 6
2.2 Probability Mass Function
2.3 Families of Discrete Random Variables
2.4 Cumulative Distribution Function (CDF)
N. Ruzgar
MTH514 W14 Lecture 7, 8
1/75
MTH514 Probability and Stochastic Processes
Lecture 7, 8
2.5 Averages
2.6 Func
.0“er
RYERSON UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH 514 - Probability and Stochastic Processes
First Term Test - Winter 2011
Last Name (print): R50
First Name (print):
Student ID Number:
Signature:
Instructor:
Escobar
Date: February 3, 2011, 2
Answer Key for Exam A
Section 1. Identify Test
1. This is the rst question of the multiple choice. You have to get this one right! Mark it on your scantron
right away. The large framed letter next to Exam Form: above is:
(a) A
(b) B
(c) C
Section 2. Multi