Lecture 20

Zhihua (Sophia) Su

University of Florida

Feb 27, 2015

STA 4321/5325 Introduction to Probability 1

Agenda

Gamma Random VariablesNormal Random Variables

Reading assignment: Chapter 4: 4.5, 4.6

STA 4321/5325 Introduction to Probability 2

Gamma Random Variables

Some random variables have the property thatP (X ∈ (x− h, x+ h)) increases as we move away from zero, andthen decreases again as we get close to large values of X. Inother words, the chance of observing very small or very largevalues is negligible, and most of the values observed are close tothe average associated with the random variable X, i.e., E(X).

STA 4321/5325 Introduction to Probability 3

Gamma Random Variables

DefinitionA random variable X is said to be a Gamma random variable if

X = Range(X) = (0,∞),

fX(x) =

{1

Γ(α)βαxα−1e−x/β if x ≥ 0,

0 if x < 0.

Hence α > 0, β > 0 are fixed parameters for the distribution.

Note that when α = 1, the Gamma density reduces to theexponential density with parameter β. Hence the exponentialrandom variable is a special case of the Gamma random variablewith α = 1.

STA 4321/5325 Introduction to Probability 4

Gamma Random Variables

The first thing that should be verified is whether the densityintegrates to 1.

STA 4321/5325 Introduction to Probability 5

Gamma Random Variables

Find the expected value E(X) and variance V (X) for a Gammarandom variable X.

STA 4321/5325 Introduction to Probability 6

Normal Random Variables

The most widely use continuous random variable in probabilityand its applications is the “normal random variable”. A randomvariable X is said to be a normal random variable withparameters µ ∈ R and σ > 0 if

X = Range(X) = R,

andfX(x) =

1√2πσ2

e−(x−µ)2

2σ2 , x ∈ R.

STA 4321/5325 Introduction to Probability 7

Normal Random Variables

What do the parameters µ and σ mean in terms of the randomvariable X?

Result

E(X) = µ, and V (X) = σ2.

Let us postpone the proof of this result for later and look at theshape of the normal density.

STA 4321/5325 Introduction to Probability 8

Normal Random Variables

The normal distribution works as a good model for a lot ofmeasurements that are observed in real experiments. There is avalid reason for this and we will see that a few weeks later, butfor now let us contend ourselves with the knowledge that if therandom quantity under consideration is an average ofindependent random quantities, then the normal distribution isquite likely an appropriate model for that random quantity.

STA 4321/5325 Introduction to Probability 9