Jointly Distributed Random Variables IE 502: Probabilistic Models - - PowerPoint PPT Presentation

Jointly Distributed Random Variables

IE 502: Probabilistic Models Jayendran Venkateswaran IE & OR

Joint Distribution Functions

• X and Y are random variables
• Joint cumulative distribution function of X and Y

– Individual CDFs of X and Y

• X & Y are discrete r.v

 joint probability mass function

– Individual pmfs

• X & Y are jointly continuous

 joint probability density function

– Individual pdfs

Examples

• 1. Two balls are drawn from box shown.

Let X be the number of red balls & Y be the number of blue balls drawn. Find joint distribution and marginal distributions of X and Y.

• 2. Suppose X and Y are jointly continuous

random variables with joint density fX,Y(x, y) = cex+y for x, y in (-∞, 0] and fX,Y(x, y) = 0, otherwise.

• What is value of c?
• What is the probability X < Y?
• What are the marginal densities fX and fY?

Properties of Expectations

• X and Y are random variables and g is a function
• f the two variables, then

in discrete case E[g(x, y)] = in continuous case

• Other properties of Expectations
• Example: Bernoulli r.v. & Binomial r.v.

∑ ∑

y x

y x p y x g ) , ( ) , (

∫ ∫

∞ ∞ − ∞ ∞ −

dxdy y x f y x g ) , ( ) , (

Variance and Covariance

• Covariance of any two random variables X and Y,

is defined as:

Cov(X, Y) =

• Properties of Covariance
• Variance of sum of random variables
• Properties of Variance

Independent Random Variables

• Random variables X and Y are said to be

independent iff, P{X ≤ a, Y ≤ b} = P{X ≤ a}·P{Y ≤ b}

• In Example (2), are X and Y independent?

Sums of Random Variables

• To determine the distribution of the sum of

independent random variables in terms of the distribution of the individual constituents

– X & Y are independent r.v – Need to find CDF of Z=X+Y – X & Y are continuous – X has density fX and Y has density fY

Sum of random variables

Calculate density of X+Y for the following cases… a) X and Y are independent r.v., both uniformly distributed on (0, 1). b) X and Y are independent Exponential r.v. with parameter λ. c) X and Y are independent Poisson r.v. with means λ1 and λ2 .