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TN 208: STOCHASTIC SIGNALS
AND SYSTEMS
Module 1:
Probability and Random Variables
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Random Experiments,
Sample Space and Sample Point,
Events, Mutually Exclusive Events,Independent Events.
Probability definition and theorems,
Random variable definition.
Classification of random variables.
To be Covered
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To be Covered
Cumulative Distribution Function
(cdf).
Probability Density Function (pdf).
Statistical Averages.
Common Probability Distributionfunctions.
Gaussian random variables.
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Probability
Probability implies random experiments.
A random experiment can have many possibleoutcomes; each outcome known as a sample point(a.k.a. elementary event) has some probability assigned.This assignment may be based on measured data orguestmates (equally likely is a convenient and oftenmade assumption).
Sample Space S : a set of all possible outcomes(elementary events) of a random experiment. Finite (e.g., if statement execution; two outcomes) Countable (e.g., number of times a while statement is
executed; countable number of outcomes)
Continuous (e.g., time to failure of a component or signal)
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Probability
Definition
A probabilistic experiment, or randomexperiment, or simply an experiment, isthe process by which an observation is
made. In probability theory, any action or process that
leads to an observation is referred to as anexperiment.
Examples include: Tossing a pair of fair coins.
Throwing a balanced die.
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Probability
Definition
The sample space associated with aprobabilistic experiment is the set
consisting of all possible outcomes of theexperiment and is denoted by S. The elements of the sample space are
referred to as sample points.
A discrete sample space is one that containseither a finite or a countable number ofdistinct sample points.
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Probability
Definition
An event in a discrete sample space Sis acollection of sample points, i.e., any subset of S.In other words, an event is a set consisting of
possible outcomes of the experiment. Definition
A simple event is an event that cannot bedecomposed. Each simple event corresponds to
one and only one sample point. Any event thatcan be decomposed into more than one simpleevent is called a compound event.
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Probability
Definition
Let A be an event connected with a probabilisticexperiment Eand let Sbe the sample space of E.
The event
Bof nonoccurrence of
Ais called thecomplementary event of A.
This means that the subset Bis the complement Aof A in S.
In an experiment, two or more events are said to beequally likely if, after taking into consideration allrelevant evidences, none can be expected inreference to another.
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Probability
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Probability
Axiomatic Approach
Analyzing the concept of equally likely probability, wesee that three conditions must hold.
1. The probability of occurrence of any event must
be greater than or equal to 0.
2. The probability of the whole sample space mustbe 1.
3. If two events are mutually exclusive, the
probability of their union is the sum of their
respective probabilities. These three fundamental concepts form the basis of
the definition of probability.
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Probability
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Probability
Let A, B and C be events in the sample space, S, then
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MUTUALLY EXCLUSIVE EVENTS
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MUTUALLY EXCLUSIVE EVENTS
Events are mutually exclusive if they cannot
happen at the same time.
For example, if we toss a coin, either heads or
tails might turn up, but not heads and tails at
the same time.
Similarly, in a single throw of a die, we can
only have one number shown at the top face.
The numbers on the face are mutually
exclusive events
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MUTUALLY EXCLUSIVE EVENTS cont..
IfA and B are mutually exclusive
events then the probability ofA
happening OR the probability ofBhappening is P(A) + P(B).
P(A or B) = P(A) + P(B)
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Example 1 What is the probability of a die showing a 2 or
a 5?
MUTUALLY EXCLUSIVE EVENTS cont..
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Practice The probabilities of three teams A, B and C
winning a badminton competition are
Calculate the probability that
a) either A or B will win
b) either A or B or C will win
c) none of these teams will win
d) neither A nor B will win
MUTUALLY EXCLUSIVE EVENTS cont..
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Solution/s
c) P(none will win) = 1 P(A or B or C will win)
d) P(neither A nor B will win) = 1 P(either A or B will win)
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Independent Events
Events are independent if the outcome of oneevent does not affect the outcome of another.
For example, if you throw a die and a coin, the
number on the die does not affect whether the
result you get on the coin.
IfA and B are independent events, then the
probability ofA happening AND the probability
ofB happening is P(A) P(B).
P(A and B) = P(A) P(B)
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Example 1
If a dice is thrown twice, find the probability
of getting two 5s.
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Two sets of cards with a letter on each card as
follows are placed into separate bags.
Sara randomly picked one card from each bag.
Find the probability that:
a) She picked the letters J and R.b) Both letters are L.
c) Both letters are vowels.
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Solution for no. 2
a) Probability that she picked J and R =
b) Probability that both letters are L =
c) Probability that both letters are vowels =
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Example 3
Two fair dice, one colored white and
one colored red, are thrown. Find
the probability that: a) the score on the red die is 2 and
white die is 5.
b) the score on the white die is 1 and
red die is even
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Solution for No. 3
a) Probability the red die shows 2
and white die 5 =
b) Probability the white die shows 1
and red die shows an evennumber =
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DEPENDENT EVENTS
Events are dependent if the outcome ofone event affects the outcome ofanother. For example, if you draw two
colored balls from a bag and the firstball is not replaced before you draw thesecond ball then the outcome of the
second draw will be affected by theoutcome of the first draw.
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DEPENDENT EVENTS cont..
IfA and B are dependent events, then
the probability ofA happening AND the
probability ofB happening, givenA, is
P(A) P(B afterA).
P(A and B) = P(A) P(B afterA)
P(B afterA) can also be written as P(B |A)
then P(A and B) = P(A) P(B |A)
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Example 1
A purse contains four P50 bills, five P100bills and three P20 bills. Two bills areselected without the first selection being
replaced.
Find P(P50, then P50)
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Solution
There are four P50 bills.
There are a total of twelve bills.
P(P50) = 4/12
The result of the first draw affected the
probability of the second draw.
There are three P50 bills left. There are a total of eleven bills left.
P(P50 after P50) = 3/11
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P(P50, then P50) = P(P50) P(P50
after P50) = (4/12)x(3/11)=12/132
The probability of drawing a P50bill and then a P50bill is
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Dependent: Practice
A bag contains 6 red, 5 blue and 4
yellow marbles. Two marbles are
drawn, but the first marble drawnis not replaced.
a) Find P(red, then blue). b) Find P(blue, then blue)
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Independent Events: Practice
Two fair dice, one colored white and
one colored red, are thrown. Find
the probability that: a) the score on the red die is 2 and
white die is 5.
b) the score on the white die is 1 and
red die is even
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Mutually Exclusive Events: Practice
The probabilities of three teams A, B and C
winning a badminton competition are
Calculate the probability that
a) either A or B will win
b) either A or B or C will win c) none of these teams will win
d) neither A nor B will win
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Summary
For mutually exclusive eventsPr(A or B) = Pr(AB) = Pr(A)+Pr(B)
For independent eventsPr(A and B)=Pr(A B) = Pr(A)Pr(B)
In general,Pr(A B) = Pr(A)+Pr(B)-Pr(A B)Pr(A B) = Pr(A)+Pr(B)-Pr(A B)
Pr(A B)=Pr(B|A)Pr(A)=Pr(A|B)Pr(B)
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Sample Space Worked Examples
Problem 1:Count the number of voice packets containing only
silence produced from a group of N speakers in a 10-ms period.
Solution: Denote sample space by S then,
S = { 0, 1, 2, , N }
Problem 2:A block is transmitted repeatedly over a noisy
channel until an error-free block arrives at the receiver. Count
the number of transmission required.
Solution: Denote sample space by S then,
S = { 1, 2, 3, , }
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Sample Space Worked Examples
Problem 3:Measure the time between two message arrivals at a
message center.
Solution: Denote sample space by S then,
S = { t: t 0} = [ 0, )
where t denotes time.
Problem 4:Measure the lifetime of a given computer memory
chip in a specified environment.
Solution: Denote sample space by S then,
S = { t: t 0} = [ 0, )
where t denotes time.
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Events Worked examples
Problem 1:Write the values of events for problems in case
study of sample space for following events:
1. No active packets are produced
2. Fewer than 10 transmission are required
3. Less than t0 seconds elapse between message arrivals
4. The chip lasts for more than 1000 hours but fewer than 5000
hour
Solution :
1. No active packets are produced, then
A = { 0 }
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Events Worked examples cont..
2.Fewer than 10 transmission are required
A = { 1, 2, , 9 }
3.Less than t0 seconds elapse between message arrivalsA = { t : 0 t < t0 } = [ 0, t0 )
4. The chip lasts for more than 1000 hours but fewer than5000 hour
A = { t : 1000 < t < 5000 } = (1000, 5000 )