Lecture 21

Zhihua (Sophia) Su

University of Florida

Mar 9, 2015

STA 4321/5325 Introduction to Probability 1

Agenda

The Standard Normal Distribution Function and ItsPropertiesExamples

Reading assignment: Chapter 4: 4.5

STA 4321/5325 Introduction to Probability 2

The Standard Normal Distribution Functionand Its Properties

DefinitionIf X is a normal random variable with parameters µ = 0 andσ = 1, then Z is said to be a standard normal random variable.

Note that, by definition, the density of a standard normalrandom variable is given by

fZ(z) =1√2πe−

z2

2 , for z ∈ R.

STA 4321/5325 Introduction to Probability 3

The Standard Normal Distribution Functionand Its Properties

Find E(Z) and V (Z).

STA 4321/5325 Introduction to Probability 4

The Standard Normal Distribution Functionand Its Properties

ResultIf Z is a standard normal random variable, then for any µ ∈ Rand σ > 0, X = µ+ σZ is a normal random variable withparameters µ and σ, i.e., the probability density function of Xis given by

fX(x) =1√

2πσ2e−

(x−µ)2

2σ2 , for x ∈ R.

Prove that E(X) = µ and V (X) = σ2.

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The Standard Normal Distribution Functionand Its PropertiesThe distribution function of a standard normal random variablehas a special name and place in the theory of probability.

DefinitionThe “Φ-function” from R to [0, 1] is defined as

Φ(z) =

ˆ z

−∞

1√2πe−

x2

2 dx = P (Z ≤ z) for z ∈ R.

Hence Φ is essentially the distribution function of a standardnormal random variable. It cannot be evaluated in closed form(except at special values), but with modern computingtechnology there are programs which will give you the value ofΦ at any given point. Let us study some of the importantproperties of this function.

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The Standard Normal Distribution Functionand Its Properties

Property 1

0 ≤ Φ(z) ≤ 1, for z ∈ R.

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The Standard Normal Distribution Functionand Its Properties

Property 2

Φ(z) + Φ(−z) = 1.

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The Standard Normal Distribution Functionand Its Properties

Property 3

If X is a normal random variable with mean µ and variance σ2,then

FX(x) = P (X ≤ x) = Φ

(x− µσ

).

STA 4321/5325 Introduction to Probability 9

Examples

Example: Sick leave time need by employees of a firm in thecourse of 1 month has approximately a normal distribution witha mean of 180 hours and a variance of 350 hours. Find theprobability that the total sick leave for a given month is lessthan 150 hours. Your answer may be expressed in terms of theΦ-function.

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Examples

Suppose that men’s neck sizes are approximately normallydistributed with a mean of 16.2 inches and variance of 0.81inches. Find the probability that the neck size of a randomlychosen man lies between 13.5 and 18.9 inches.

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