Probability and Statistics - Lecture Slides | ECE 596A, Study notes of Electrical and Electronics Engineering

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ECE 596-C: Network Simulation

Probability & Statistics

Basic concepts

Independent events

Random variable

Cumulative distribution function (CDF)

Probability density function(PDF)

Probability mass function

Mean or expected value

Variance

Coefficient of variation

Ratio of standard deviation to mean

Covariance

Cov(X, Y) = E[ (X - E(X)) (Y - E(Y)) ] = E[XY] - E[X] E[Y]

Summarizing data by a single number

Indices of central tendencies

Mean, median, and mode

Some characteristics

Mean exists and is unique; Mode may not exist or may not be unique

Outliers may drastically affect mean

Mean has linearity property, while mode and median do not

Selecting among mean, median, and mode

If data is categorical, use mode

If total is of interest, use mean

If distribution is skewed, use median

Otherwise, use mean!

Common misuse of means

Using means of significantly different values

Using means without regard to skewness of distribution

Simple way to measure skewness is the ratio of maximum to minimum

Multiplying means to get the mean of product

Taking mean of a ratio with different bases

Defining mean of ratios

Example

CPU busy times measured over several intervals

Compute the mean busy duration

Approximates to arithmetic and harmonic means

If the denominator is constant: arithmetic mean

If the numerator is constant: harmonic mean

Mean of ratios

Measurement Duration CPU Busy (%) 1 45 1 45 1 45 1 45 100 20 Sum 200% Mean 40%?

Mean of ratios

If numerator and denominator follows multiplicative property

a(i) = c. b(i)

c may be estimated by geometric mean of a(i)/b(i)

Example

Benchmarking a program optimizer

Program Code SizeCode Size Program Ratio Before After Ratio P1 119 89 0. P2 158 134 0. P3 142 121 0. P4 8612 7579 0. P5 7133 7062 0. P6 184 112 0. P7 2908 2879 0. P8 433 307 0. Geometric MeanGeometric MeanGeometric Mean 0.

Summarizing variability

Quartiles

25-, 50-, 75-percentiles

Q1 = 25-percentile, Q2 = 50-percentile, Q3 = 75-percentile

Semi-Interquartile Range (SIQR)

Mean absolute deviation

How are these measures affected by outliers?

Selecting index of dispersion

Is distribution bounded?

If yes, use range

Is distribution unimodal symmetrical?

If yes, use coefficient of variance

Otherwise, use percentiles or SIQR

Confidence interval for the mean

Confidence interval computed as

: -quantile of a unit normal variate

Above applies to samples of greater than 30

For smaller samples, use t-variate with (n-1) degrees of freedom

Testing for zero mean

Check if the measured value is significantly different from 0

Same procedure may be applied to other values

Mean 0

Approximate visual test

Simpler method is to compare two unpaired samples

Perform the t-test only in the third scenario

Mean Mean^ Mean

Other considerations

What confidence level to use?

Hypothesis testing vs. confidence intervals

One-sided confidence intervals

Sample size to use for determining mean with a desired confidence interval