Random Variables

Dr. Abdulaziz Almulhem

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Preliminary

An important design issue of networking is the ability to model and estimate performance parameters

For example, estimate future traffic volumes and characteristics

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Why do we need such estimates?

To study the effect of routing protocols

To estimate resources needed by reservation protocols

To study queuing discipline

To identify buffer sizes needed

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Preliminary

Parameters used in characterizing data traffic:

Throughput characteristics:1. Average rate: the load sustained by the

source over a time period (resource allocation)

2. Peak rate: the max. load a source can generate (buffering might be needed for smoothing)

3. Variability: burstiness of a source

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Preliminary

Delay characteristics:1. Transfer delay: delay from source to

destination

2. Delay variation (jitter): variation in transfer delay (impacts real-time applications)

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What’s next?

We need to know little about probability

Random variablesWhat are they?Their propertiesExamples

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Probability Premier

Probability P(A) of an event A is a number that corresponds to the likelihood that the event A will occur

Sample Space (space of events)

A

B

0 1P(A) P(B)

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Definitions & observations

0 P(A) 1

P(Ai) = 1; Ai is an event in the sample space

P(A)= Na/N;Na= number of outcomes in which A occurred

(frequency)N= total number of possible outcome

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Definitions & observations

If two events A & B are mutually exclusive (independent) then:Prob (A or B is to occur) =P(A) + P(B)Prob (A and B to occur) =P(A) * P(B)

EX. Out of 2 apples and 3 oranges in a basket, what is the prob. of having 2 oranges when I need to grab three items from the basket?

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Definitions & observations

The conditional prob. of an event A assuming the event B has occurred P(A|B) is (A & B are not independent):P(A|B)=P(AB)/P(B) If A & B are independent:

P(A|B)=P(A) & P(A|B)=P(B)

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Baye’s Theorem

Given the set of mutual exclusive events E1, …, En

Ei covers an arbitrary event A

P(A)=in

=1 P(A|Ei)P(Ei)=?

Then

P(Ei|A)=P(A|Ei)P(Ei)/P(A)E3

E1

E2

A

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Example

Given,

S0 = event of sending 0

S1 = event of sending 1

R0 = event of receiving 0

R1 = event of receiving 1

P(S0) = p P(S1) = 1-p

Also the received data (bits) can be observed

P(R0|S1) = pa & P(R1|S0) = pb

Physical Medium

Network

Sender Receiver

0 0

1 1

Error

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Example

Now to calculate the conditional probability of an error

That is a one was sent given that a zero is received

P(S1|R0)= P(R0|S1) P(S1) / P(R0)

Where P(R0) = P(R0|S0) P(S0) + P(R0|S1) P(S1)

P(S1|R0)=pa p / [pa p+(1-pb)(1-p)]

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Random Variables

RV is simply a numerical description of the outcome of a random experiment.Examples:

Arriving customers at a given time Tossing a coinPackets in a switch at a given timeEtc.

We describe RV with distribution functions.

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Cumulative distribution function (CDF)

CDF for an RV denoted FX(x) is defined as the probability that RV is less than or equal to x:

FX(x) = p(X x)

F(-)=0; F()=1; 0F(x) 1

F(x1) F(x2) when x1 x2

p(x1 X x2) = F(x2) - F(x1)

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Probability distribution function (pdf)

It is the derivative of CDF

fX(x) = d FX(x) / dx

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Moments

To completely characterize a RV, it is sufficient to know its pdf.

It is practical to describe some key aspects or few numbers of the pdf rather than specifying the entire pdf.

This is called moments or statistical avergaresEvaluated using the mathematical expectation

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Mathematical expectation

The expected or mean value of an RV

Expectation is a linear operation

dxxxf

px

xFEx

kxk

xx)(

xx xFE )(

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Moments

The mth-order moment of FX(x) is

Zero order =1

First order is mean (previous slide)

Second is the mean-squared value

dxxfxxFE xmm

x )()(

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Moments (cont.)

Second moment is the variance and denoted by

Standard deviation is square root of variance and it measures the speard of observed values of the RV around its mean

2x

xxxx xFExF )()(var 2

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Moments (cont.)

Third moment describes the skewness and characterizes the degree of asymmetry of the distribution around its mean.

It is a dimensionless quantity. When zero distribution is symmetric +ve leans towards the right; -ve leans towards the

left

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Moments (cont.)

Fourth moments defines kurtosis and measures the flatness or peakedness of a distribution about its mean.

It is dimensionless It is relative to the normal distribution More +ve means peaked distribution More –ve means flatten distribution