Lecture 24

Zhihua (Sophia) Su

University of Florida

Mar 16, 2015

STA 4321/5325 Introduction to Probability 1

Agenda

Joint Probability Distributions for Discrete RandomVariables

Reading assignment: Chapter 5: 5.1, 5.2

STA 4321/5325 Introduction to Probability 2

Joint Probability Distributions for DiscreteRandom Variables

Until now we have been studying situations, where we are onlyinterested in one random quantity (or random variable) arisingout of an experiment. But there are various situations, wherethere is more than one quantity associated with the experiment,and we are interested in the JOINT behavior of these randomquantities. Let us consider a simple motivating example.

STA 4321/5325 Introduction to Probability 3

Joint Probability Distributions for DiscreteRandom Variables

On April 15, 1912, the ocean liner Titanic collided with aniceberg and sank. Of the 2201 passengers on board, 1496perished. The question as to whether passenger class wasrelated to survival has been discussed extensively. Here is atable summarizing the details of number of survivors by class.

Passenger Status Survivors Fatalities TotalFirst class 203 122 325Second class 118 167 285Third class 178 528 706

Crew 212 673 885Total 711 1490 2201

STA 4321/5325 Introduction to Probability 4

Joint Probability Distributions for DiscreteRandom VariablesThere are two discrete random variables in play here,

X =

{0 if passenger survived,1 if passenger did not survive.

Y =

1 if passenger was in first class,2 if passenger was in second class,3 if passenger was in third class,4 if passenger was a crew member.

Experiment: Select a Titanic passenger randomly. What is thechance that he/she is a non-crew member who survived? Toanswer questions like these, we develop a concept of the jointprobability mass function of two discrete random variables.

STA 4321/5325 Introduction to Probability 5

Joint Probability Distributions for DiscreteRandom Variables

DefinitionLet X, Y be discrete random variable. Let X =Range(X),Y =Range(Y ). The joint probability mass function of (X,Y ) isdefined as

pX,Y (x, y) = P (X = x, Y = y) for x ∈X , y ∈ Y .

STA 4321/5325 Introduction to Probability 6

Joint Probability Distributions for DiscreteRandom VariablesIn the same fashion, we define the concept of a joint probabilitydistribution function of two random variables X and Y .

DefinitionLet X and Y be two random variables arising out of anexperiment. Then the joint probability distributionfunction of X and Y is defined as

FX,Y (a, b) = P (X ≤ a, Y ≤ b) for any a, b ∈ R.

If X, Y are discrete, then

FX,Y (a, b) =∑

x∈X :x≤a

∑y∈Y :y≤b

P (X = x, Y = y).

STA 4321/5325 Introduction to Probability 7

Joint Probability Distributions for DiscreteRandom Variables1 lima→−∞ limb→−∞ FX,Y (a, b) = P (X ≤ −∞, Y ≤ −∞) = 0.

2 lima→∞ limb→∞ FX,Y (a, b) = P (X ≤ ∞, Y ≤ ∞) = 1.

3

P (a < X ≤ b, c < Y ≤ d)

= P (X ≤ b, Y ≤ d)− P (X ≤ a, Y ≤ d)− P (X ≤ b, Y ≤ c)

+P (X ≤ a, Y ≤ c)

= FX,Y (b, d)− FX,Y (a, d)− FX,Y (b, c) + FX,Y (a, c)

4 For every fixed a, the function FX,Y is right continuous inthe second coordinate. For every fixed b, the function FX,Y

is right continuous in the first coordinate.

It can be proven that these properties characterize a probabilitydistribution function.

STA 4321/5325 Introduction to Probability 8

Joint Probability Distributions for DiscreteRandom Variables

Getting back to the example, find the joint probability massfunction of X and Y .

STA 4321/5325 Introduction to Probability 9

Joint Probability Distributions for DiscreteRandom Variables

Using the joint probability mass function of X and Y , we cancalculate the individual or marginal probability mass functionsof X and Y .

ResultIf X and Y are discrete random variable with joint massfunction pX,Y , then the individual or marginal probability massfunction of X and Y are given by

pX(x) = P (X = x) =∑y∈Y

P (X = x, Y = y) =∑y∈Y

pX,Y (x, y),

pY (y) = P (Y = y) =∑x∈X

P (X = x, Y = y) =∑x∈X

pX,Y (x, y).

STA 4321/5325 Introduction to Probability 10

Joint Probability Distributions for DiscreteRandom Variables

As an example, suppose we are interested in the probability thata randomly chosen passenger is a survivor.

STA 4321/5325 Introduction to Probability 11

Joint Probability Distributions for DiscreteRandom VariablesHowever, many times we need to calculate conditionalprobabilities of events related to the random variables X and Y .

ResultLet X and Y be two discrete random variable. The conditionalprobability mass function of X given Y = y is defined as

pX|Y=y(x) = P (X = x | Y = y) =P (X = x, Y = y)

P (Y = y).

Similarly, the conditional probability mass function of Y givenX = x is defined as

pY |X=x(y) = P (Y = y | X = x) =P (X = x, Y = y)

P (X = x).

STA 4321/5325 Introduction to Probability 12

Joint Probability Distributions for DiscreteRandom Variables

In the Titanic example, what is the probability that a passengersurvived given that he/she was a first class passenger?

STA 4321/5325 Introduction to Probability 13