1
Exam 2 Version A
1. Solution:
a) We must have:
1
cx(1 2x)dx = c
0
Therefore, c = 6.
b) FX (x) =
x
0
1
0
2
x 2x2 dx = c( x
2
2
6y(1 2y)dy = 6( x
2
2x3 1
3 )|0
2x3
3
1
= c( 2
2
3
0) = c( 1 ) = 1
6
CMPE 107 Spring 2016
Sample Exam 2 Solutions
Name: Awesome Student-Seeking-A+
YOU MUST JUSTIFY ALL YOUR ANSWERS!
1. (15 points) Show that the mean [or variance] of the Poisson random variable equals t
1
Exam 1- A
1- show (A - C)(C - B)=
Solution:
(A Cc )(C Bc ) = (C Cc )(A Bc ) = (A Bc ) =
2- If 4 socks are randomly drawn from a bag containing 6 blues and 5 red
socks what is the probability that y
CMPE 107 Spring 2016
Take-Home Exam
MUST BE SUBMITTED IN HARD COPY BY JUNE 7 NO LATER THAN 4PM [in JJs office
or with Ali in E2 315]
Grading system: presentation = 25%, content = 75%
Last Name:
First
Probability and Statistics
for Engineers
(CMPE 107)
!
http:/users.soe.ucsc.edu/~jj/CLASSES/CMPE107-Spring16/
J.J. Garcia-Luna-Aceves
University of California
Santa Cruz
(UCSC)
[email protected]
http:/us
my
CMPE 107 Quiz 1
1. Define a random experiment and its sample space
4. Consider two events A and B defined on some sample space Q, and let their respective
probabilities be 0.4 and 0.7. Are these
CMPE 107 Winter 2015
Two Single-Sided Pages of Notes Allowed
Exam 3A
Name:
YOU MUST JUSTIFY ALL YOUR ANSWERS TO RECEIVE ANY CREDT!
1. (20 points) Random variable X and Y have the following joint PDF:!
CMPE 107 Spring 2016
Sample Exam 1
Name: Awesome Student-Seeking-A+
YOU MUST JUSTIFY ALL YOUR ANSWERS!
1. (20 points) Let A, B and C be sets. Use set identities to show (A C) (C B) =
(A Cc) (C B c) =
CMPE 107 1st Quiz
Solutions
1. Consider a computer that generates random binary words with 8 bits. Assume that all
words have the same probability of being generated.
What is the probability that the
CMPE 107 Spring 2016
Sample Exam 1
Name: Awesome Student-Seeking-A+
YOU MUST JUSTIFY ALL YOUR ANSWERS!
1. (20 points) Let A, B and C be sets. Use set identities to show (A C) (C B) =
2. (20 points) I
CMPE 107 Spring 2016
Exam 3 Sample
Name: You The following are sample questions to show you the type of questions you can
expect. All questions are from HWs and class notes. You already have the answe
Probability as Frequency of Occurrence: The probability of an outcome equals the relative frequency with which it
occurs if an experiment is repeated a very large number of times under the same condit
CMPE 107 - Homework 7
Solutions
NOTE: If a RV is uniform with parameters a and c, it means that it is uniformly
distributed over the interval [a-c/2,a+c/2]
1. The voltage x across a 1 resistor is a un
CMPE 107 - Homework 5
due Feb. 17
1. Consider a box with 10 balls, 4 of which are red and the others are white. Suppose
you are drawing 5 balls at random. Consider the random variable x, where cfw_x=n
CMPE 107 - Homework 4
1. Consider a sequence of binary words with 5 bits. After measuring the relative
frequencies of 0s and 1s in the binary words, we estimate that the probability that a
bit in a wo
CMPE 107 - Homework 8
due Mar. 13
Exercise 1
Suppose the pdf fx(x) of a random variable x is such that fx(a+x)=fx(a-x) for some value a.
Prove that E[x]=a.
Prove that the mean and the median of x co
CMPE 107 - Homework 4
1. Consider a sequence of binary words with 5 bits. After measuring the relative
frequencies of 0s and 1s in the binary words, we estimate that the probability that a
bit in a wo
Chapter
1
M/M/1 QUEUE
Im sure that Ive never been in a queue as slow as this.
Any Customer, Anywhere, Anytime
Nobody goes there anymore. Its too crowded.
Yogi Berra
The M/M/1 queue, the simplest and m
HOMEWORK 8 - SOLUTIONS
Exercise 1
a
"
E[x ] =
"
# xf ( x ) dx = # xf (x ) dx + # xf (x ) dx =
x
x
!"
x
a
!"
0
0
0
# (x + a) f ( x + a) dx + # (a ! x) f (a ! x ) dx = 2a # f ( x + a ) dx =
=
x
x
!"
!"
CMPE 107 1st HOMEWORK (due Jan. 20)
1. How many three digit numbers can be formed from the digits 1,2,3,4,5 and 6, if each
digit can only be used once? How many of these are odd numbers? How many are
CMPE 107 2nd HOMEWORK (due Jan. 27)
1. Consider a computer that uses binary words with 8 bits.
How many words exist such that the number of bits equal to 1 is at least twice the
number of bits equal t
CMPE 107 - Homework 6
due Feb. 27
1. The random variable x is normal, having pdf equal to g ,! ( x ) with =2.
a. What is P(x 2)?
b. Knowing that P(x > 4)=0.16, can you compute P(x 0)?
c. Knowing that
Politecnico di Torino
Network Modeling
Formulary
Fiandrino Claudio - Pievanelli Elisa
Academic Year 2010/2011
Contents
I
Theory
5
1 Elementary Queueing Theory
1.1 Introduction and Basic Definitions .
CMPE 107 - Homework 9
Exercise 1
Given a random variable x that has an exponential pdf with parameter , find the 90%
confidence interval, i.e., the interval [0,T] such that P(0<xT = 0.9).
Exercise 2
G
CMPE 107 Winter 2017 Exam 1
Name/Student ID:
YOU MUST JUSTIFY ALL YOUR ANSWERS!
l. (20 points) Let A, B and C be three events in a sample space. Let P(A) = 122, P(B) = US, and P(C) = U4
(a) Use set id
CMPE 107 Spring 2017 CLOSED BOOK, CLOSED NOTES Exam 2
Last Name: First Name: ID:
YOU MUST JUSTIFY ALL YOUR ANSWERS TO RECEIVE ANY CREDT!
l. (15 points) In CMPE 107, we know that It) percent of the stu
CMPE 107 Spring 2017
CLOSED BOOK, CLOSED NOTES
Last Name:
First Name:
Exam 3
Student ID:
1. (20 points) Random variable X and Y have the following joint PDF with a being a constant
"$ 10a 0 y x 1,
f X
CMPE 107 Spring 2017 Exam 1
Last Name First Name _ Student ID:
YOU MUST JUSTIF Y ALL YOUR ANSWERS!
1. (20 points) Let A and B be two events that need not be mutually exclusive. Let P(A) = 1/3 and P(B)