Lecture 26

Zhihua (Sophia) Su

University of Florida

Mar 20, 2015

STA 4321/5325 Introduction to Probability 1

Agenda

Independent Random Variables

Reading assignment: Chapter 5: 5.4

STA 4321/5325 Introduction to Probability 2

Independent Random Variables

Let us recollect the notion of “independent events”. We say thatevents A and B are independent if

P (A ∩B) = P (A)P (B) or P (A | B) = P (A).

The way we understand it intuitively is that even if we are giventhe information that B has occurred, that does not change theprobability of A. The same notion can be generalized to randomvariables. Let us consider the case of discrete random variables.

STA 4321/5325 Introduction to Probability 3

Independent Random Variables

DefinitionLet X, Y be discrete random variables. Then X and Y are saidto be independent if

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

for every x ∈ X , y ∈ Y .

STA 4321/5325 Introduction to Probability 4

Independent Random Variables

Note that

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

is the same as

P (X = x | Y = y) = P (X = x) or P (Y = y | X = x) = P (Y = y).

Hence, saying that X and Y are independent means that even ifwe are given the information that Y = y, that does not changethe probability behavior of X. Similarly, even if we are giventhe information that X = x, that does not change theprobability behavior of Y .

STA 4321/5325 Introduction to Probability 5

Independent Random Variables

Let us now turn our attention to continuous random variables.

DefinitionLet X, Y be continuous random variables. Then X and Y aresaid to be independent if

fX,Y (x, y) = fX(x)fY (y),

for every x ∈ R, y ∈ R.

STA 4321/5325 Introduction to Probability 6

Independent Random Variables

Note thatfX,Y (x, y) = fX(x)fY (y)

is the same as

fX|Y=y(x) = fX(x) or fY |X=x(y) = fY (y).

Hence, saying that X and Y are independent, implies that theprobability behavior of X is unaffected by information about Y ,and the probability behavior of Y is unaffected by informationabout X.

STA 4321/5325 Introduction to Probability 7

Independent Random Variables

Example: A bus arrives at a bus stop at a randomly selectedtime within a 1-hour period. A passenger arrives at the bus stopat a randomly selected time with the same hour. The passengeris willing to wait for the bus for up to 1/4 of an hour. What isthe probability that the passenger will catch the bus?

STA 4321/5325 Introduction to Probability 8