Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.5 PROPERTIES OF DISTRIBUTIONS

In many circumstances, however, random variables do not depend on one

another, i.e. they areindependent. As an example, for a person drawn at random

from a population, we might expect height and IQ to be independent random

variables. Let us suppose thatXandYare two random variables with probability

density functionsg(x)andh(y) respectively. In mathematical terms,XandYare

independent RVs if their joint probability density function is given byf(x, y)=

g(x)h(y). Thus, for independent RVs, ifXandYare both discrete then

Pr(X=xi,Y=yj)=g(xi)h(yj)

or, ifXandYare both continuous, then

Pr(x<X≤x+dx, y < Y≤y+dy)=g(x)h(y)dx dy.

The important point in each case is that the RHS is simply the product of the

individual probability density functions (compare with the expression for Pr(A∩B)

in (30.22) for statistically independent eventsAandB). By a simple extension,

one may also consider the case where one of the random variables is discrete and

the other continuous. The above discussion may also be trivially extended to any

number of independent RVsXi,i=1, 2 ,...,N.

The independent random variablesXandYhave the PDFsg(x)=e−xandh(y)=2e−^2 y

respectively. Calculate the probability thatXlies in the interval 1 <X≤ 2 andYlies in

the interval 0 <Y≤ 1.

SinceXandYare independent RVs, the required probability is given by

Pr(1<X≤ 2 , 0 <Y≤1) =

∫ 2

1

g(x)dx

∫ 1

0

h(y)dy

=

∫ 2

1

e−xdx

∫ 1

0

2 e−^2 ydy

=

[

−e−x

] 2

1 ×

[

−e−^2 y

] 1

0 =0.^23 ×^0 .86 = 0.^20 .

30.5 Properties of distributions

For a single random variableX, the probability density functionf(x) contains

all possible information about how the variable is distributed. However, for the

purposes of comparison, it is conventional and useful to characterisef(x)by

certain of its properties. Most of these standard properties are defined in terms

ofaveragesorexpectation values. In the most general case, the expectation value

E[g(X)] of any functiong(X) of the random variableXis defined as

E[g(X)] =

{∑

ig(xi)f(xi) for a discrete distribution,

∫

g(x)f(x)dx for a continuous distribution,

(30.45)

where the sum or integral is over all allowed values ofX. It is assumed that