Probability for Engineers
Random Variables and probability distributions
Lecture 04
Sajid Bashir, PhD
sajid@iqraisb.edu.pk
Outline
Random Variables
Probability Distributions
Probability Distribution Function
Probability Mass Function
Probability Density F
Probability for Engineers
Instructor
Sajid Bashir, PhD
COURSE LOGISTICS
Sajid Bashir,PhD
Random Variables and Probability Distributions
Sep15
2
Course Logistics
Course Book
T. T. Soong, Fundamentals of Probability and Statistics for
Engineers.
Roy D. Yat
Methods for Dummies 2009
Bayes for Beginners
Georgina Torbet & Raphael Kaplan
Bayesian Probability
Probability: often used to refer to frequency
but
Bayesian Probability: a measure of a state of
knowledge.
It quantifies uncertainty. Allows us to reason u
Probability for Engineers
Random Variables and probability distributions
Lecture 05
Sajid Bashir, PhD
sajid@iqraisb.edu.pk
Outline
Random Variables
Probability Distributions
Probability Distribution Function
Probability Mass Function
Probability Density F
Mathematical Methods
Probability & Random Variables
Lecture 01
Sajid Bashir, PhD
sajid@iqraisb.edu.pk
COURSE LOGISTICS
Sajid Bashir,PhD
Random Variables and Probability Distributions
Sep15
2
Course Logistics
About me
Instructor
Dr Sajid Bashir
Contact
sa
Probability for Engineers
Expectation & Moments
Lecture 06
Sajid Bashir, PhD
sajid@iqraisb.edu.pk
Outline
Moments of a Single Random Variable
Mean, Median, and Mode
Central Moments, Variance, and Standard Deviation
Conditional Expectation
Chebyshev Inequa
Mathematical Methods
Probability & Random Variables
Lecture 03
Sajid Bashir, PhD
sajid@iqraisb.edu.pk
Probability & Random Variables
LEC 02 RECAP
Sajid Bashir,PhD
Random Variables and Probability Distributions
Sep15
2
Set Theory and Probability
Sajid Bas
Name :
Score :
Teacher :
Date :
Solve the Problems Using the Venn Diagrams
A
B
4
2
13
1
12
3
9
15
19
20 5
11
C
1)
(A U C) B
= _
2)
(A U B) C
= _
3)
C  (A B)
= _
4)
C U (A  B)
= _
5)
A  (B C)
= _
6)
(A C)  B
= _
7)
AUBUC
= _
8)
B U (A C)
= _
9)
C U (B
www.MathWorksheetsGo.com
On Twitter: twitter.com/engagingmath
On FaceBook: www.mathworksheetsgo.com/facebook
I. Model Problems.
II. Practice
III. Challenge Problems
IV. Answer Key
Web Resources
Complex Number Lessons, Links and Videos
www.mathwarehouse.co
Name :
Score :
Teacher :
Date :
Name the Shaded Regions for the Venn Diagrams
1 ) Region _
A
B
2 ) Region _
A
B
C
C
3 ) Region _
4 ) Region _
A
B
A
B
C
C
5 ) Region _
6 ) Region _
A
B
C
A
B
C
MathAids.Com
Name :
Score :
Teacher :
Date :
Name the Shaded R
Name :
Score :
Teacher :
Date :
Answer the Questions Based on the Venn Diagram
Swimming
Skiing
5
7
19
13
6
8
10
9
Boxing
1 ) How many students like both Swimming, Skiing, and Boxing ? _
2 ) How many students do not like both Swimming and Boxing ? _
3 ) Ho
Subject: Math
Unit: Number Concepts
Lesson: Two
email:info@youtheducationservices.ca
www.youtheducationservices.ca
The Rational Number System
Worksheet
Classify these numbers as rational or irrational and give your reason.
1.
a. 7329
b. 4
2.
a. 0.95832758
CCAF Challenge
College Mathematics
 Consists of 60 questions and given 90 minutes
 Approx 50% will be straightforward math questions and
other 50% word problems
 This lesson is to jar your memory and to get you to pass
the College Mathematics CLEP
 So
Sets of Numbers in the Real Number System
Reals
A real number is either a rational
number or an irrational number.
2
4, 7, 0, , 11
3
Rationals
A rational number is any number that
p
can be put in the form
where p
q
and q are integers and q z 0 .
12 5 1 8
CCAF Challenge
Complex Numbers  Logarithms  Geometrical Shapes  Algebra
Complex Numbers
(Module 5)
Complex numbers are represented by a + bi, where a and b are real numbers and i is an imaginary unit.
a+bi, a is called the real part of the complex num
Complex Conjugates
Every complex number has a complex conjugate. The complex conjugate of a + bi is a  bi . For example, the
conjugate of 3 + 15i is 3  15i , and the conjugate of 5  6i is 5 + 6i .
When two complex conjugates a + bi and a  bi are added
studs
ACADEMIC
o
Coll
\
\.
O The natural numbers are the set ofnumbers used when
counting; these are the numbers cfw_ 1, 2, 3, 4, 5, i.
o The whole numbers are the set of natural numbers and
zero; these are the numbers cfw_0, l, 2, 3, 4, 5, l.
O The i
Easy Permutations and Combinations  BetterExplained
Page 1 of 95
Learn Right, Not Rote.
Easy Permutations and Combinations
Ive always confused permutation and combination which ones which?
Heres an easy way to remember: permutation sounds complicated, do
Set Theory Symbols and Definitions
Symbol
Name
Definition
Example
Set
A collection of elements
A = cfw_2,7,8,9,15,23,35
Intersection
Objects that belong to set A and set B
If set A = cfw_1,2,3 & set B = cfw_2,3,4
then
= cfw_2,3
Union
Objects that belong t
Worksheet 2.7 Logarithms and Exponentials
Section 1 Logarithms
The mathematics of logarithms and exponentials occurs naturally in many branches of science.
It is very important in solving problems related to growth and decay. The growth and decay
may be t
PERMUTATIONS AND COMBINATIONS WORKSHEET
CTQR 150
1. If the NCAA has applications from 6 universities for hosting its intercollegiate tennis championships in 2008 and 2009, how many ways may they select the hosts for these championships
a) if they are not
Arguments and Truth Tables
Definition:
Deductive reasoning proceeds step by step from assumed statements called premises or
hypotheses to a statement called a conclusion.
Definition:
A valid argument is an argument in which the conclusion must be true whe
Domain and Range
Several of the Math IC questions about functions will focus on domain and range.
These questions are straightforward if you understand the basic concepts and know
what to look for.
Domain
We discussed the concept of a functions domain ear
Number properties, Rational numbers, Irrational numbers, Math worksheet for Grade 7 at . Page 1 of 2
Name: _
Grade: 7
Date: 12/18/2013
Instructions: Answer the following questions.
Question 1
A rational number can always be written in
what form?
A.
decima
Functions and their Graphs  Complex Numbers  Logarithms
(Module 4)
Brain Melt Edition
Roughly 12 questions will be about functions and their graphs.
A function is a relationship between two quantities (x, y), where one value depends on the relationship
Probability and Statistics
(Module 3)
Roughly 1215 questions on the CLEP test involve probability and statistics. Probability and statistics include
Counting, Range, Standard Deviation, Tables, Bar Graphs, etcetcetc.
Probability is the likelihood that so
Sets
Module 2

Union and intersection
Subsets, disjoint sets, equivalent sets
Venn diagrams
Cartesian product
Sets, Subsets, Disjoint Sets and Equivalent Sets
Simply put, a set is a collection of objects and those objects are referred to as elements of t
Complex Numbers, Log and Exponents, Geometrical Shapes, Fundamental Theorem of
Algebra
(Module 5)
Complex numbers are represented by a + bi, where a and b are real numbers and i is an imaginary unit. In the
definition a+bi, a is called the real part of th
Lesson Plan
College Mathematics
(Overview)
The college mathematics consists of 60 questions that you must complete within 90 minutes. The goal
of this lesson plan is to jar your memory and get you to pass the test. I am not a math expert but I have
resear