Lecture 7

Zhihua (Sophia) Su

University of Florida

Jan 23, 2015

STA 4321/5325 Introduction to Probability 1

Agenda

Properties of distribution function

STA 4321/5325 Introduction to Probability 2

Properties of distribution function

Let us recollect that if X is a random variable, then itsdistribution function FX : R→ [0, 1] is defined by

FX(b) = P (X ≤ b) for all b ∈ R.

STA 4321/5325 Introduction to Probability 3

Properties of distribution function

Property I

limb→−∞

FX(b) = 0.

STA 4321/5325 Introduction to Probability 4

Properties of distribution function

Property II

limb→∞

FX(b) = 1.

STA 4321/5325 Introduction to Probability 5

Properties of distribution function

Property III

FX is a non-decreasing function.

STA 4321/5325 Introduction to Probability 6

Properties of distribution function

Property IV

FX is right-hand continuous, i.e.,

limh→0+

FX(b+ h) = FX(b).

However, it is not true that FX is left-hand continuous,especially for discrete random variables.

STA 4321/5325 Introduction to Probability 7

Properties of distribution function

Example: Let X be the number of heads for 3 tosses of a faircoin. Find and plot the distribution function FX .

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Properties of distribution function

Fact: Any function satisfying properties I, II, III and IV is adistribution function of some random variable.

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