Lecture 6

Zhihua (Sophia) Su

University of Florida

Jan 21, 2015

STA 4321/5325 Introduction to Probability 1

Agenda

Random VariablesProbability Mass FunctionProbability Distribution Function

Reading assignment: Chapter 2: 2.11-2.13,Chapter 3: 3.1-3.2

STA 4321/5325 Introduction to Probability 2

Random Variables

Many times when we observe a random experiment, there is anumerical quantity associated with the experiment that we areinterested in. For example,

(i) Experiment: Sample n people (without replacement) froma population of N people.Quantity of interest: The average height.

(ii) Experiment: A pandemic spreading through town.Quantity of interest: Percentage of people affected.

(iii) Experiment: Toss a coin 3 times.Quantity of interest: # of heads.

These quantities of interest associated with random experimentsare called random variables, generally denoted by X, Y , Z, . . ..

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Random Variables

DefinitionA random variable is a real-valued function whose domain is thesample space.

Notationally, X : S → R.

The set of possible values that a random variable X takes, orequivalently, the range of X is generally denoted by X .

DefinitionA random variable is called discrete if the set of possible valuesthat it takes is discrete.

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Probability Mass Function

We focus on discrete random variables for a while. It is quiteimportant to know the various chances with which the randomvariable takes various values. Mathematically, we need to knowP (X = x) for every x ∈X .

DefinitionThe probability mass function of a random variable X(discrete), is given by

pX(x) ≡ P (X = x) for every x ∈X .

Note that pX(x) ≥ 0 and∑

x∈X pX(x) = 1.

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Probability Mass Function

Example 1: Sample n people (without replacement) from apopulation of N people, with no preference to any specificpeople. The quantity of interest (X) is the average height.Suppose N = 3, n = 2, and we have 3 people with heights say50, 60, 70 inches.

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Probability Mass Function

Example 2: Toss a fair coin 3 times. X = # of heads (quantityof interest).

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Probability Distribution FunctionSometimes it is important to study a random variable bylooking at its cumulative properties, i.e., probabilities of thetype P (X ≤ b) for any real number b.

DefinitionA distribution function F for a random variable X is defined by

FX(b) ≡ P (X ≤ b) for all b in R (real number).

If X is a discrete random variable,

FX(b) = P (X ≤ b) =∑

x∈X :x≤bpX(x).

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Probability Distribution Function

Find the distribution function for the random variable X inExample 1.

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