Lecture 25

Zhihua (Sophia) Su

University of Florida

Mar 18, 2015

STA 4321/5325 Introduction to Probability 1

Agenda

Joint Distribution Functions for Continuous RandomVariablesExamples

Reading assignment: Chapter 5: 5.2, 5.3

STA 4321/5325 Introduction to Probability 2

Joint Probability Distributions for ContinuousRandom Variables

We previous studied the joint probability mass functions forjointly describing the probability behavior of two discreterandom variables. Today, we repeat the same exercise forcontinuous random variables.

STA 4321/5325 Introduction to Probability 3

Joint Probability Distributions for ContinuousRandom Variables

Let us recollect that a random variable X is said to becontinuous, if it has a density function fX , such that

(i) fX(x) ≥ 0 for every x ∈ R, fX(x) = 0 for everyx 6∈X =Range(X).

(ii) P (a ≤ X ≤ b) =´ ba fX(x)dx.

STA 4321/5325 Introduction to Probability 4

Joint Probability Distributions for ContinuousRandom Variables

DefinitionLet X and Y be two continuous random variables. The jointprobability behavior of X and Y is described by the jointprobability density function fX,Y which has the followingproperties.(i) fX,Y (x, y) ≥ 0 for every (x, y) ∈ R2.

(ii) P (a ≤ X ≤ b, c ≤ Y ≤ d) =´ dc

´ ba fX,Y (x, y)dxdy.

It follows that the joint probability distribution function FX,Y isgiven by

FX,Y (a, b) = P (X ≤ a, Y ≤ b) =

ˆ b

−∞

ˆ a

−∞fX,Y (x, y)dxdy.

STA 4321/5325 Introduction to Probability 5

Joint Probability Distributions for ContinuousRandom Variables

Example: A certain process for producing an industrial chemicalyields a product that contains two main types of impurities. LetX denote the proportion of impurities of Type I and Y denotethe proportion of impurities of Type II. Suppose that the jointdensity of X and Y can be adequately modeled by the followingfunction:

fX,Y (x, y) =

{2(1− x) 0 ≤ x ≤ 1, 0 ≤ y ≤ 1,

0 otherwise,

Compute P (0 ≤ X ≤ 0.5, 0.4 ≤ Y ≤ 0.7).

STA 4321/5325 Introduction to Probability 6

Joint Probability Distributions for ContinuousRandom Variables

DefinitionIf X and Y are continuous random variables with jointprobability density function fX,Y , then the individual ormarginal density functions fX and fY are given by the following:

fX(x) =

ˆ ∞−∞

fX,Y (x, y)dy for every x ∈ R,

fY (y) =

ˆ ∞−∞

fX,Y (x, y)dx for every y ∈ R.

STA 4321/5325 Introduction to Probability 7

Joint Probability Distributions for ContinuousRandom Variables

Example: In the industrial production example consideredpreviously, compute fX and fY .

STA 4321/5325 Introduction to Probability 8

Joint Probability Distributions for ContinuousRandom Variables

Suppose we are interested in the behavior of the randomvariable X given that Y = y. To develop a framework forexpressing this behavior, we need the notion of “conditionalprobability density function”.

STA 4321/5325 Introduction to Probability 9

Joint Probability Distributions for ContinuousRandom VariablesDefinitionLet X and Y be continuous random variables with jointprobability density function fX,Y and marginal densities fX andfY . Then the conditional probability density function of Xgiven Y = y is defined by:

fX|Y=y(x) =

{fX,Y (x,y)fY (y) y ∈ Y = Range(Y ),

0 otherwise.

Similarly, the conditional probability density function of Ygiven X = x is defined by

fY |X=x(y) =

{fX,Y (x,y)fX(x) x ∈X = Range(X),

0 otherwise.

STA 4321/5325 Introduction to Probability 10

Joint Probability Distributions for ContinuousRandom Variables

Example: Compute the conditional probability density functionsfX|Y=y and fY |X=x for the industrial impurities example.

STA 4321/5325 Introduction to Probability 11