View
227
Download
0
Embed Size (px)
Citation preview
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}
123456
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,
dx
xdFxf
)()( =
∫=−=≤<2
1
)()()()( 1221
x
x
dxxfxFxFxxxP
Expected Value (Mean)
• Discrete random variable
• Continuous random variable
∑ ===valuespossibleall
xXPxXE )(.)(μ
∫+∞
∞−== dxxxfXE )()(μ
Variance / Standard Deviation…
• Variance for a random variable X
• Standard deviation is :• Coefficient of variation: • Covariance:
• Coefficient of correlation:
€
Var (X) =σ X2 = E (x −μ)2
[ ] = E(X 2) − E(X)2
€
σX
€
S tandard Deviation
Mean=σ
μ
€
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)