Random Variables, Distributions, and Moments IE 502: Probabilistic - - PowerPoint PPT Presentation

Random Variables, Distributions, and Moments

IE 502: Probabilistic Models Jayendran Venkateswaran IE & OR

IE502: Probabilistic Models IEOR @ IITBombay

Example: ‘Continuum’

• Suppose a student has a class at 9.30am. The student, as

usual leaves his hostel in the last minute. He is likely to be late but will definitely reach the class before 10.30am. How likely is it that student is delayed by no more than 30 mins?

• Outcome

Random variable

• Wall Clock Time

delay time, X, [0, 1] hour

• To obtain probabilities, let’s assume an auxiliary function, f, to

be −2x + 2 for x in [0, 1]

IE502: Probabilistic Models IEOR @ IITBombay

Continuous r.v. – density function

• Continuous r.v.  whose set of possible values

are uncountable.

– Makes definition of pdf more cumbersome than pmf

• Probability density function (pdf) assigns a

probability density to each and every value of r.v.

• Example: ‘Continuum’ Delay

IE502: Probabilistic Models IEOR @ IITBombay

Summary

• Random variables

– Defining r.v: Discrete and continuous – Outcomes, random variables, and events

• Cumulative distribution function (cdf): F(x)
• Discrete r.v.

– Probability mass function (pmf): p(x)

• Continuous random variable

– Probability density function (pdf): f(x)

IE502: Probabilistic Models IEOR @ IITBombay

Expectation of random variable

• Expectation or Expected Value or Mean
• 1. If X is a discrete r.v. with pmf p(x), then expected

value of X is …

• Three coin toss example
• 1. If X is a continuous r.v. with pdf f(x), then

expected value of X is …

– Delay example

IE502: Probabilistic Models IEOR @ IITBombay

Expectation of a Function of a r.v.

• We are interested in the expected value of some

function of r.v. X, say, g(X)

• Proposition 2.1

– If X is a discrete random variable with pmf p(x), then for any real-values function g, – If X is a continuous random variable with pdf f(x), then for any real-values function g,

∑

=

x

x p x g X g E ) ( ) ( )] ( [

∫

∞ ∞ −

= dx x f x g X g E ) ( ) ( )] ( [

IE502: Probabilistic Models IEOR @ IITBombay

Moments of a random variable

• The nth moment of r.v. X is E[Xn], n≥1

– 1st moment is the mean

• The nth central moment of r.v. X is E[ (X − E[X])n ]

– First central moment is 0 – Second central moment is Variance, the amount of variability in the values of the r.v. – Positive square root of variance is standard deviation σ

• Compute Var(X)

– Three coin toss example – Student delay example

IE502: Probabilistic Models IEOR @ IITBombay

Central Moments

• Normalized moments

– Normalized nth moment is the nth central moment divided by σn

• Third central moment

– Measures lopsidedness of the distribution – Skewness normalized third central moment – Skewness = 0 for symmetric distributions

Negative skew Positive skew

IE502: Probabilistic Models IEOR @ IITBombay

Central moment (contd)

• Fourth Central Moment

– Measure of whether distribution is measure of “peakedness” – distribution is tall and skinny or short and squat (compared to the Normal distribution) – Kurtosis  normalized fourth central moment − 3

Theoretical Distributions

IE 502: Probabilistic Models Jayendran Venkateswaran IE & OR

IE502: Probabilistic Models IEOR @ IITBombay

Introduction

• Theoretical distributions have been derived based
• n certain facts/ principles/ assumptions by logical
• r mathematical reasoning

– Allows us to make rational judgments concerning probability of events

IE502: Probabilistic Models IEOR @ IITBombay

Discrete Probability Distributions

IE502: Probabilistic Models IEOR @ IITBombay

Bernoulli Distribution

• Named after James Bernoulli

– Bernoulli trials

• X ~ Bernoulli(p)
• X takes on value 1 with success

probability p and value 0 with failure probability q=1-p

• PMF
• Expected Value & Variance

Bernoulli trial is an experiment whose

• utcome is random &

can be either of 2 possible outcomes, "success" & "failure"

IE502: Probabilistic Models IEOR @ IITBombay

Binomial Distribution

• Has foundation in Bernoulli trials/process
• X ~ Binomial(n, p)
• X is the number of successes in n trials, where

each trial can result in success with probability p and failure with probability q = 1-p

• PMF
• Expected Value & Variance

IE502: Probabilistic Models IEOR @ IITBombay

Bernoulli trials & Binomial Distribution

IE502: Probabilistic Models IEOR @ IITBombay

Binomial Distribution - pmf

p=0.5, n =10 p=0.5, n =20 p=0.7, n =20

IE502: Probabilistic Models IEOR @ IITBombay

Geometric Distribution

• Has foundation in the Bernoulli trials/process
• X ~ Geometric(p)
• X is the number of trials required until the first

success, where each trial can result in success with probability p

• PMF
• Expected Value & Variance