1. Summarizing Measured Data Part I Visualization (Chap 10) Part II Data Summary (Chap 12)

2. 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)

3. A Good Graphic Chart Requires minimum effort from reader Maximizes information Minimizes ink: (crowded versus information) Uses commonly accepted practices Avoid ambiguity

4. 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..

5. Histograms Qualitative data on x-axis Percentage TCP version On Internet TCP Tahoe TCP RenoNewRenoOther

6. 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?

7. 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%

8. 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

9. 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 LU Th LR ADT

10. Part II Data Summary Chap. 12

11. Probability In networks particularly, experiments are subject to uncertainty, to variability. Probability provides means to characterize uncertainty.

12. 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

13. 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

14. 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}

15. 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)

16. 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

17. Probability Distributions DISCRETE 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} 123456123456 0 1 Bernouilli random variable P(0) = P(X = 0) =? P(1) = P(X = 1) = ?

18. 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

19. Probability Distributions CONTINUOUS 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,

20. Expected Value (Mean) Discrete random variable Continuous random variable

21. Variance / Standard Deviation… Variance for a random variable X Standard deviation is : Coefficient of variation: Covariance: Coefficient of correlation:

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

23. Discrete Distributions Binomial probability distribution Hypergeometric probability distribution Negative binomial distribution Poisson probability distribution

24. 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

25. Moments Definition: the k th moment of a distribution f(x) is E(X k ). Examples: –First moment is E(X) (Mean) –Second moment is E(X 2 )…(Handy to get the variance)