EE250
Probability, Random
Variables and
Stochastic Processes
Instructor: Dr. Juzi Zhao
Fall 2022
Lecture 1:
Course Overview
Probability Models
Lecture outline
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Course overview
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Instructor information
Course materials
Probability Models
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Course Information
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Lectures:
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8/19/2022
TuTh 3:00PM - 4:15PM
Zoom link:https://sjsu.zoom.us/j/85980439008
This course will be recorded for
instructional/educational purposes. The
recordings will only be shared with students
enrolled in the class through Canvas. The
recordings will be deleted at the end of the
semester
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Primary Instructor: Dr. Juzi Zhao
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E-mail: juzi.zhao@sjsu.edu
Office hours:
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Wednesday Office Hour 9:00-10:00am, Zoom link:
https://sjsu.zoom.us/j/85478022485
Friday Office Hour 4:00-5:00pm, Zoom link:
https://sjsu.zoom.us/j/86131063875
Available by appointment at other times
ISA: TBD
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Course materials
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Textbook:
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Probability and Random Processes for Electrical
Engineering by A. Leon-Garcia, Prentice Hall, (3rd
Edition). Required.
Course materials such as syllabus, handouts, notes,
assignments, etc. can be found on Canvas Leaning
Management System course login website at
http://sjsu.instructure.com.
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Course policies
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Grading breakdown
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Midterm 1 (Sept 22)
Midterm 2 (Nov. 3)
Final exam (Dec. 9)
Assignments
25 %
25 %
30 %
20 %
Online open book open notes exams
might use a Proctoring service such as
ProctorU
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Exams
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The final exam date is fixed. If you have 3 or more final
exams in 24 hours, you can request to take a make-up
final exam on Thursday, Dec. 15. Please email me to
schedule the make-up exam time at least 2 weeks
before Dec. 15.
If you request to re-schedule the midterm exams, please
let me know (by email) at least 2 weeks before the
corresponding midterm exam dates (Sept. 22 and Nov.
3, respectively). I will ask the rest of class about their
availability and preference.
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Bonus
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There may be a short group quiz during some classes to
check whether or not you understand some concepts
covered in the class
Break the class randomly into small groups
Group members have discussions on the quiz questions
(via Zoom breakout rooms)
Randomly choose one group to share/explain their
solutions to the class
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0.5 point as bonus to each member of the selected group if their
solutions are correct
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Sets
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A set is a collection of unique objects
Here “objects” can be concrete things (people in class,
schools in San Jose), or abstract things (numbers, colors)
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We use capital letters to denote sets
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The things that together make up the set are elements
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Sets
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Two basic ways to specify a set:
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1. List all the elements, separated by commas, inside a pair of
braces: A={0, 1, 2, 3}
2. Give a property that specifies the elements of the set:
A={x: x is an integer such that 0≤x≤3}
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Building Sets Using Conditionals
Mathematical rule for generating all of the elements of
the set: perform the operation to the left of the vertical
bar on the numbers to the right of the bar
C  {x 2 | x  1,2,3,4,5}
D  {x 2 | x  1,2,3,...}
Is 10 an element of set D? Is 144 an element
of set D? Is 49 an element of set C?
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Sets
Order in a set does not matter!
{1, 2, 3} = {3, 1, 2} = {1, 3, 2}
We use small letters to denote elements
When x is an element of A, we denote this by: x ∈ A
If y is not in a set A, we denote this as:
y
A
The “empty” r “null” set has no elements:
∅= { }
Universal set U is the set of all elements of interest
in a given setting or application
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Subsets
A set A is a subset of another set B if every
element of A is also an element of B, and we
denote this as A  B. Every set is a subset of U
 Examples {1, 9}  {1, 3, 9, 11}
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{ apple, pear}  { apple, pear, banana}
∅  A for any set A
A={x: x≥10} C={x:x ≥ 20}
Set equality A = B
A = B if and only if A
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B and B
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A
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Set Operations:Complement
Definition
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The complement of a set A U, denoted Ac, is the set
of all elements in U that are not in A.
Ac = {x|x ∈ U, xA}
Example: roll a six-sided die and observe the number of dots
on the side facing upwards
A = {1, 3, 5}
“an odd roll”
The complement of U is null set ∅
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Set Operations: Union
Definition
The union of two sets A and B, denoted A ∪ B is the set
of all elements in either A or B (or both).
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A ∪ B = {x|x ∈ A or x ∈ B or both}
Example: roll a six-sided die and observe the number of
dots on the side facing upwards
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A = {1, 3, 5}
B = {1, 2, 3}
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“an odd roll”
“a roll of 3 or less”
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Set Operations:Intersection
Definition
The intersection of two sets A and B, denoted A ∩ B is
the set of all elements in both A and B.
A ∩ B = {x|x ∈ A and x ∈ B}
Example: roll a six-sided die and observe the number of dots
on the side facing upwards
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A = {1, 3, 5} “an odd roll”
B = {1, 2, 3} “a roll of 3 or less”
Note: If A ∩ B = ∅, we say A and B are disjoint or
mutually exclusive.
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Set Operations: Difference
Definition
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The difference of a set A U and a set B U,
denoted A − B, is the set of all elements in U that are in A
and are not in B.
A - B = {x|x ∈ A and x
B}
Example:
A = { 3, 4, 5, 6}
B = { 3, 5}
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Visualize Set Operations with Venn
Diagrams
U
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Examples
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A={x: x≥10} B={2,4,6,...} (assume
elements are integers)
Find A ∩ B, A ∪ B, A-B, B-A, Ac , Bc
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Properties
Commutative properties:
A ∩ B= B ∩ A, A ∪ B= B ∪ A
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Associative properties:
A ∩ (B ∩ C)= (A ∩ B) ∩ C
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A ∪ (B ∪ C) = (A ∪ B) ∪ C
Distributive properties:
A ∪ (B ∩ C)= (A ∪ B) ∩ (A ∪ C)
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A ∩ (B ∪ C)= (A ∩ B) ∪ (A ∩ C)
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DeMorgan’s Law
(A ∪ B)c = Ac ∩ Bc
(A ∩ B)c = Ac ∪ Bc
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Examples
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A={v: |v| > 10}, B={v: v<-5}, C={v: v>0}
(assume elements are integers)
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A ∩ B, A ∪ B,Cc,(A ∪ B) ∩ C,A ∩ B ∩ C,(A ∪
B)c
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Random Experiment
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A random experiment to be an experiment
in which the outcome varies in an
unpredictable fashion when the experiment is
repeated under the same conditions
Random experiment is specified by stating an
experimental procedure and a set of one or
more measurements or observations
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Examples
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Select a ball from an urn containing balls numbered 1 to 50. Note
the number of the ball.
Toss a coin three times and note the sequence of heads and tails.
Toss a coin three times and note the number of heads.
A block of information is transmitted repeatedly over a noisy channel
until an error-free block arrives at the receiver. Count the number of
transmissions required.
Pick a number at random between zero and one.
Measure the lifetime of a given computer memory chip in a specified
environment.
Determine the values of an audio signal at times T1 and T2
Pick two numbers at random between zero and one.
Pick a number X at random between zero and one, then pick a
number Y at random between zero and X.
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Outcome and Sample Space
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An outcome or sample point of a random experiment
as a result that cannot be decomposed into other results
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When we perform a random experiment, one and only
one outcome occurs
outcomes are mutually exclusive
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The sample space S of a random experiment is defined
as the set of all possible outcomes
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Each performance of a random experiment can then be
viewed as the selection at random of a single point
(outcome) from S
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Sample space
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A sample space can be finite, countably
infinite, or uncountably infinite
S a discrete sample space if S is countable
S a continuous sample space if S is not
countable
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Events
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An event is a subset of S
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The event occurs if and only if the outcome of the experiment is in
this subset
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Two events of special interest are the certain event, S, which
consists of all outcomes and hence always occurs, and the
impossible or null event ∅, which contains no outcomes and
hence never occurs
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An event may consist of a single outcome
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An event from a discrete sample space that consists of a single
outcome is called an elementary event.
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Event Class
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A class of events is a collection (set) of
events (sets)
If the class C consists of the collection of sets
A1, A2, A3,... Ak then C={A1, A2, A3,... Ak }
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