Random Variables and
Stochastic Processes – 0903720
Dr. Ghazi Al Sukkar
Email: ghazi.alsukkar@ju.edu.jo
Office Hours: will be posted soon
Course Website:
http://www2.ju.edu.jo/sites/academic/ghazi.alsukkar
Most of the material in these slide are based on slides prepared by Dr. Huseyin Bilgekul
http://faraday.ee.emu.edu.tr/ee571/
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PART 1
Axioms of Probability
1. Set review
1.1 Introduction
1.2 Sample Space and Events
1.3 Axioms of Probability
1.4 Basic Theorems
1.5 Continuity of Probability Function
1.6 Probabilities 0 and 1
1.7 Random Selection of Points from Intervals
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Set Review I
• A set { } is a collection of elements. Sets are denoted by
CAPITAL letters, and elements by small letters.
• The symbol ∈ means “is an element of”, and ∉ means “is
not an element of”.
• The symbol ∅ denotes the empty set.
• A set is countable if it is finite, or it is infinite and its
elements has a one-to-one correspondence with the
integers.
• Uncountable set is an infinite set which is not countable.
• A set A is contained in a set B (A is proper “strict” subset
of B), denoted A ⊂ B ( or B ⊃ A), if every element of A is
also in B.
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Set Review II
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𝐴 ⊆ 𝐵 : subset has fewer elements or equal to the set.
𝐴 𝑜𝑟 (𝐴𝑐 ) complement: all the objects that do not belong to set A.
A = B if and only if A ⊂ B and B ⊂ A.
(𝐴 ∖ 𝐵) 𝑜𝑟 (𝐴 − 𝐵)relative complement: objects that belong to A
and not to B.
𝐴 × 𝐵 Cartesian product: set of all ordered pairs from A and B.
The union of two sets A and B, denote A ∪ B, is the set of all
elements that belongs to A or B or both.
The intersection of two sets A and B, denote A ∩ B, is the set of
elements that belong to both A and B.
𝐴 △ 𝐵 𝑜𝑟 (𝐴 ⊝ 𝐵)symmetric difference: objects that belong to A
or B but not to their intersection.
𝐴 cardinality: the number of elements of a finite countable set A.
For more: http://www.rapidtables.com/math/symbols/Set_Symbols.htm
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1.1: Introduction
Overview:
An event may or may not occur
• What’s Random?
Occurrence of an event is inevitable
• What’s Certain?
An event can never occur
• What’s Impossible?
Examples：
1. Disintegration of a given atom of radium
2. Finding no defect during inspection of a microwave oven
3. Orbit satellite in space is at a certain position
4. An object travels faster than light
5. A thunderstorm flashes of lighting precede any thunder echoes
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Random experiment
• Random experiment: its outcome, for some
reason, cannot be predicted with certainty.
• Examples: throwing a die, flipping a coin and
drawing a card from a deck.
Relative Frequency Interpretation
• Definition： (Probability based on experiment): The number of times the
experiment is performed is n, and nA is the number of times, where the outcome
belongs to A ⊂ S.
nA
The probability P of event A is defined as
P ( A)  lim
n 
Dilemmas：
n
1.
2.
3.
Can not be computed since n, only approximation
Does the limit of nA/n exist?
Probabilities that are based on our belief and knowledge are not justifiable.
The probability that the price of oil will be raised in
the next six months is 60%.
The probability that it will snow next Christmas is 30%
• Classical Definition: N is the total number of outcomes, and NA is the number of
outcomes that belongs to A ⊂ S.
P(A) =NA/N
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1.2: Sample Space and Events
• Sample Space：
The set of all possible outcomes, denoted by S
note: A sample space can be discrete or continuous.
 Sample points：
These outcomes are called sample points, or points.
Outcomes are denoted by 𝜁’s and each 𝜁 lies in , i.e., 𝜁 ∈ S.
 Events：
Certain subsets of S are referred to as events.
Sample Space
b
a
d
k
f
e
c
h
g
j
m
l
i
n
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Sample point
Event
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Examples
 Example 1.1：
For the experiment of tossing a coin once, what is the
sample space S ?
 Example 1.2：
Suppose that an experiment consists of two steps.
First a coin is flipped. If the outcome is tails, a die is
tossed. If the outcome is head, the coin is flipped again.
What is the sample space S ? What is the event of
heads in the first flip of the coin? What is the event
of an odd outcome when the die is tossed?
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Examples
 Example 1.3：
Consider measuring the lifetime of a light bulb. Since
any nonnegative real number can be considered as the
lifetime of the light bulb (in hours), the sample space S
is S = {x：x  0 }.
The event E = {x：x  100 } is the event that the light
bulb lasts at least 100 hours.
The event F = {x：x  1000 } is the event that it lasts at
most 1000 hours.
The event G = {505.5} is the event that it lasts exactly
505.5 hours.
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Examples
 Example 1.4：
Suppose that a study is being done on all families with
one, two, or three children. Let the outcome of the
study be the genders of the children in descending
order of their ages. What is the sample space S ?
What is the event F where the eldest child is a boy?
What is the event G where families with exactly 2 girls?
Ans：S = {
F={
G={
}
}
}
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Examples
 Example 1.5：
A bus with a capacity of 34 passengers stops at a
station some time between 11:00 AM and 11:40 AM
every day.
What is the sample space of the experiment, consists of
counting the number of passengers on the bus and
measuring the arrival time of the bus?
Ans:
What is the event?
F  {( 27, t ) :11 13  t  11 23}
Ans:
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Examples
 Example 1.6：(Round-off Error)
Suppose that each time Jay charges an item to his
credit card , he will round the amount to the nearest
dollar in his records. Therefore, the round-off error,
which is the true value charged minus the amount
recorded, is random, with the sample space
S={
}
The event of rounding off at most 3 cents in a random
charge is given by
S={
}
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Occurrence of an Event
 If the outcome of an experiment belongs to
an event E, we say that the event E has
occurred.
If we draw two cards from an ordinary deck of 52 cards
and observe that one is a spade and the other a heart,
all of the following events have occurred.
{sh}, {sh, dd}, {cc, dh, sh}, {hc, sh, ss, hh}, {cc, hh, sh, dd}
But none of the events {dh, sc}, {dd}, {cc, hh, ss},
{hd, hc, dc, sc, sd} have occurred.
Spade
Heart
Diamond
Club
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