Summarizing Measured Data

Part I Visualization(Chap 10)

Part II Data Summary (Chap 12)

Types of Variables

• Qualitative variables:– Finite set of values, classes: (e.g., LAN, MAN, WAN)– Ordered, or unordered

• Quantitative variables:– Numerical values– Discrete: value from a finite or countably infinite set

(number of nodes in wireless network).– Countinuous: value for an interval of real numbers

(throughput, propagation delay)

A Good Graphic Chart

• Requires minimum effort from reader

• Maximizes information

• Minimizes ink: (crowded versus information)

• Uses commonly accepted practices

• Avoid ambiguity

Mistakes to Avoid

• Too many curves

• Multiple y-variables (variables of different nature, eg., link utilization, throughput, delay, jitter…)

• Using symbols instead of plain text

• Too much detail

• Improper scale ranges

• Lines instead of columns..

Histograms

• Qualitative data on x-axis

PercentageTCP version On Internet

TCP Tahoe TCP Reno NewReno Other

Histograms (2)

• Quantitative data on x-axis

• Exercise: bottleneck bandwith is estimated at TCP senders for 1200 different paths. For each path, we have an average estimate of the bandwidth.– A) What variable should be on the y-axis?– B) What should be the intervals on the x-axis?

Gantt Chart

• Visualize the relative duration of boolean conditions

• Example 1: Processes running on CPU

• Example 2:

CPU

I/O

Network

20% 40% 60% 80% 100%

Kiviat Graph

• Circular graph representing 2n variables plotted along 2n radial lines.

• In general: – n HB (High is Better) variables on upper half– n LB (Low is Better) variables on lower half

Kiviat Graph (Example)

• A LAN is evaluated through measurement:– Link utilization is 80%– Throughput is 80 Mbps– Packet loss rate is 2%– Average delivery time is 2 ms

LUTh

LRADT

Part II

Data Summary

Chap. 12

Probability

• In networks particularly, experiments are subject to uncertainty, to variability.

• Probability provides means to characterize uncertainty.

Basics

• An experiment yields a random outcome (unique and indivisible result).

• Example: – Experiment: throw a dice– Outcome: number on upper side of dice

• An event A is a set of outcomes:– A={1,4,6}– A={x/ x is even} An experiment yields a random

outcome (unique and indivisible result).• P(A) is the probability of occurrence of event A

Basics

• An experiment yields a random outcome (unique and indivisible result).

• Example: – Experiment: throw a dice– Outcome: number on upper side of dice

• An event A is a set of outcomes:– A={1,4,6}– A={x/ x is even} An experiment yields a random outcome

(unique and indivisible result).• P(A) is the probability of occurrence of event A• Sample space S is the set of all possible outcomes for an

experiment

Probability Axioms

• 1) P(A) is positive or nul for all events A

• 2) P(S) = 1 where S is the sample space

• 3) If events A, B, C… are mutually exclusive then– P(A U B U C U..) = P(A) + P(B) + P( C) …

• Example:– Experiment= Throw a dice– Sample space S= {1, 2, 3, 4, 5, 6}

Events

• Key: an event is a SET subject to all operations on sets:– Intersection– Union– Complement

• Independent events– Two events A and B are independent if the occurrence

of A (resp. B) has no impact on the “odds” of B (resp. A) to occur.

– Formally, A and B are independent if and only if • P(A and B) = P(A).P(B)

Random Variable

• Random variable: a mapping that associates a number to each outcome in the sample space S.

• A random variable could be discrete (takes an integer value) or continuous (takes a real value).

• Examples:– Experiment: flip a fair coin. Let X be the number of trials before we

get a head– Experiment: send a packet on a channel that corrupts/loose

packets with probability p. Let X be the number of transmissions before a packet is successfully received.

– Experiment: failure of a network. Le X be the time between two successive failures of a network

Probability DistributionsDISCRETE Random Variable

• The Probability distribution or probability mass function (p.m.f) of a discrete random variable is defined for every number x by:– P(x) = P(X = x) = P(all s in S/ X(s) = x)

• Examples:– Experiment: roll a dice. Let X be 0 when we get an even number

and 1 otherwise. Sample space is {1, 2, 3, 4, 5, 6}

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0

1

Bernouilli random variable

P(0) = P(X = 0) =?P(1) = P(X = 1) = ?

Cumulative Distribution Function

• The cumulative distribution function (c.d.f) F(x) of a discrete random variable with p.m.f p(x) is defined for every number x by

– F(x) = P(X ≤ x) = p(y) for all y≤x

Probability DistributionsCONTINUOUS Random Variable

• The cumulative distribution function (c.d.f) F(x) of a continuous random variable is defined for every number x by– F(a) = P(X ≤ a)

• The Probability density function (p.d.f) of a continuous random variable is defined for every number x by:

• As a result,

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Expected Value (Mean)

• Discrete random variable

• Continuous random variable

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∫+∞

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Variance / Standard Deviation…

• Variance for a random variable X

• Standard deviation is :• Coefficient of variation: • Covariance:

• Coefficient of correlation:

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Var (X) =σ X2 = E (x −μ)2

[ ] = E(X 2) − E(X)2

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σX

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S tandard Deviation

Mean=σ

μ

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Cov(X,Y ) =σ XY2 = E (x −μX )(y −μY[ ] = E(XY ) − E(X)E(Y )

YX

XYYXσσ

σρ2

),( =

Discrete Distributions

• What to know?– Meaning/Interpretation– P(X = i)– Cumulative distribution function (P(X<=i)– Expectation– Variance

Discrete Distributions

• Binomial probability distribution

• Hypergeometric probability distribution

• Negative binomial distribution

• Poisson probability distribution

Continuous Distributions

• Less intuitive and hardly related to specific experiements (e.g, X= number of failures before a success…)

• Will detail key distributions in chapter 3

Moments

• Definition: the kth moment of a distribution f(x) is E(Xk).

• Examples:– First moment is E(X) (Mean)– Second moment is E(X2)…(Handy to get the

variance)