II. Characterization of Random Variables

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II. Characterization of Random Variables
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Random Variable

• Characterizes a random experiment in terms of
  real numbers
• Discrete Random Variables
• The random variable can only take a finite number
  of values
• Continuous Random Variables
• The random variable can take a continuum of values

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Probability Mass Function

• Only Suitable to characterize discrete random
  variables

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Cumulative Distribution Function
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Probability Density Function

• Used to characterize Continuous Random Variables

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Uniform Random Variable
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Gaussian Random Variable

• Many physical phenomenon can be modeled as
  Gaussian Random Variables most popular to
  communication engineers is AWGN Channels

mean
standard deviation
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Exponential Random Variable

• Commonly encountered in the study of queuing
  systems

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How to Characterize a Distribution
Client Tell me how good is your
network? Salesman Well, P(Delaylt1)0.1,
P(Delaylt2)0.3, P(Delaylt3)0.2, Client
Hmmm So what does this really mean? Salesman
How can I explain this?
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Mean of Random Variables
Client Tell me how good is your
network? Salesman Well, The average delay per
packet is 1 sec Client Hmmm So what does this
really mean? Salesman If you need to send 100
packets, they will most likely take 100 seconds
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Mean of a Random Variable

• Discrete Random Variable

Continuous Random Variable
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Example

• Consider a Network where the delay D is either
  1 or 5 seconds
• i.e., PD 1 0.3, PD 5 0.7
• PD 0, 2, 3, 4 0, PD 6, 7, 8, 9,
  0
• What is the mean delay?
• Let assume 100 packets, then most likely
• 30 packets will be delayed for 1 sec
• 70 packets will be delayed for 5 sec
• Therefore 100 packets will most likely take
  30x170x5 380 sec
• Average Delay 380/100 3.8 sec
• ED 1xPD15xPD5 3.8 sec

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Moments of a Random Variable

• Discrete Random Variable

Continuous Random Variable
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Central Moments

• Discrete Random Variable

Continuous Random Variable
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Variance

• Variance is a measure of random variables
  randomness around its mean value

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Conditional CDF

• Define FXAx as the conditional cumulative
  distribution function of the random variable X
  conditioned on the occurrence of the event A, then

Remember Bayess Rule
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Conditional CDF Example

• Consider a uniformly distributed random variable
  X with CDF

FXx
1
x
0
1
Calculate the conditional CDF of X given that
Xlt1/2. In other words we would like to compute
FXXlt1/2x
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Conditional CDF Example

• Consider a uniformly distributed random variable
  X with CDF

FXx
1
x
0
1
Calculate the conditional CDF of X given that
Xlt1/2. In other words we would like to compute
FXXlt1/2x
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Conditional CDF Example

• Consider a uniformly distributed random variable
  X with CDF

FXx
1
x
0
1
Calculate the conditional CDF of X given that
Xlt1/2. In other words we would like to compute
FXXlt1/2x
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Conditional CDF Example

• Consider a uniformly distributed random variable
  X with CDF

FXx
1
x
0
1
Calculate the conditional CDF of X given that
Xlt1/2. In other words we would like to compute
FXXlt1/2x
FXx
1
x
0
1/2
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Exercise

• For some random variable X and given constants
  a, b such that altb

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Conditional PDF

• Define fXAx as the conditional probability
  density function of the random variable X
  conditioned on the occurrence of the event A, then

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Conditional PDF Example

• Consider a uniformly distributed random variable
  X with CDF

fXx
1
x
0
1
Calculate the conditional PDF of X given that
Xlt1/2. In other words we would like to compute
fXXlt1/2x
fXXlt1/2x
2
1
x
1/2
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Exercise

• For some random variable X and given constants
  a, b such that altb

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Conditioning on a Characteristic of Experiment

• Conditioning does not necessarily have to be on
  the numerical outcome of an experiment
• It is possible to have qualitative conditioning
  based on a characteristic of an experiment
• Example Consider a random variable X that
  represents the score of students in a given
  course
• Conditioning based on experiment outcome
• The distribution of grades given it is greater
  than 80 (i.e., FXXgt80x)
• Conditioning based on experiment characteristic
• The distribution of grades given the gender of
  students (i.e., FXMx)

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Conditioning on a Characteristic of Experiment

• Consider a set of N mutually exclusive events
  A1, A2,, AN. Suppose we know FXAnx for n1,
  2, , N. Then

The unconditional CDF/PDF is basically the
conditioned CDF averaged across the probability
of occurrence of conditioning events Example Fo
r a bit b sent over a communication channel and
the received voltage r Prlt0Prlt0b1Pb1P
rlt0b0Pb0
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Conditioning on a Characteristic of Experiment

• Consider a set of N mutually exclusive events
  A1, A2,, AN. Suppose we know FXAnx for n1,
  2, , N. Then

Discrete Random Variable
For a continuous random variable P XxAn0, P
Xx0 resulting in an undetermined expression
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Conditioning on a Characteristic of Experiment

• Consider a set of N mutually exclusive events
  A1, A2,, AN. Suppose we know FXAnx for n1,
  2, , N. Then for a continuous random variable

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Conditional Expected Value

• The expected value of a random variable X
  conditioned on some event A

Discrete Random Variable
Continuous Random Variable