Lecture 16

Zhihua (Sophia) Su

University of Florida

Feb 13, 2015

STA 4321/5325 Introduction to Probability 1

Agenda

Properties of Probability Distribution FunctionsExamples

Reading assignment: Chapter 4: 4.2

STA 4321/5325 Introduction to Probability 2

Properties of Probability DistributionFunctionsWe saw that continuous random variables cannot be describedin terms of probability mass functions as this leads toinconsistencies with the three basic axioms of probability. As asolution, the concept of a probability density function forcontinuo random variables was introduced.

DefinitionIf X is a continuous random variable, then there exists afunction fX such that(i) fX(x) ≥ 0 for all x ∈ R (fX(x) = 0 for x 6∈X =Range(X)).(ii)

´∞−∞ fX(x)dx = 1.

(iii) P (a ≤ X ≤ b) =´ ba fX(x)dx for all a < b ∈ R.

STA 4321/5325 Introduction to Probability 3

Properties of Probability DistributionFunctions

We start by looking at some properties of the probabilitydensity function.

Hence, the ratio of the probability density function at twopoints can be roughly interpreted as the ratio of theprobabilities that X is close to the two points respectively.

STA 4321/5325 Introduction to Probability 4

Properties of Probability DistributionFunctions

If X us a continuous random variable with density fX , then

P (X = x) = P (x ≤ X ≤ x) =

ˆ X

xfX(y)dy = 0.

Hence, we inherently assign P (X = x) = 0 for every x.

Confused?Remember that we are in a reasonable situation and are tryingto come up with a reasonable framework for describingcontinuous random variables probabilistically. The frameworkwe have suggested is indeed the best possible way out.

STA 4321/5325 Introduction to Probability 5

Properties of Probability DistributionFunctions

1 Note that X can take uncountably many values. Hencesaying P (X = x) = 0 does not rule out x as a possiblevalue (at least that is not now we should interpret it).

2 Note that as long as x ∈X =Range(X),

P (x− h ≤ X ≤ x+ h) =

ˆ x+h

x−hfX(y)dy > 0.

Hence, for continuous random variables, any interval (in therange of X) is assigned a positive probability, but every point inthe range of X is assigned zero probability.

STA 4321/5325 Introduction to Probability 6

Examples

Example 1: The distribution function of the random variable X,the time (in months) from the diagnosis age until death for apopulation of patients with AIDS, is as follows

FX(x) =

{1− e−0.03x

1.2x ≥ 0,

0 x < 0.

(a) Find the probability density function of X.

STA 4321/5325 Introduction to Probability 7

Examples

(b) Find the probability that a randomly selected personsurvives at least 12 months.

STA 4321/5325 Introduction to Probability 8

Examples

Example 2: Suppose that a random variable X has a probabilitydensity function given by

fX(x) =

{x2

3 −1 < x < 2,

0 otherwise.

(a) Find the probability that −1 < X < 1.

STA 4321/5325 Introduction to Probability 9

Examples

(b) Find the distribution function of X.

STA 4321/5325 Introduction to Probability 10

Examples

Example 3: Suppose that the weekly repair cost for a certainmachine (in units of $100) denoted by the random variable X,has probability density function given by

fX(x) =

{cx(1− x) 0 ≤ x ≤ 1,

0 otherwise.

Find c.

STA 4321/5325 Introduction to Probability 11