Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

A random variableXhas a PDFf(x)given byAe−xin the interval 0 <x<∞and zero

elsewhere. Find the value of the constantAand hence calculate the probability thatXlies

in the interval 1 <X≤ 2.

We require the integral off(x) between 0 and∞to equal unity. Evaluating this integral,
we find ∫
∞

0

Ae−xdx=

[

−Ae−x

]∞

0 =A,

and henceA= 1. From (30.42), we then obtain

Pr(1<X≤2) =

∫ 2

1

f(x)dx=

∫ 2

1

e−xdx=−e−^2 −(−e−^1 )=0. 23 .

It is worth mentioning here that adiscreteRV can in fact be treated as

continuous and assigned a corresponding probability density function. IfXis a

discrete RV that takes only the valuesx 1 ,x 2 ,...,xnwith probabilitiesp 1 ,p 2 ,...,pn

then we may describeXas a continuous RV with PDF

f(x)=

∑n

i=1

piδ(x−xi), (30.44)

whereδ(x) is the Dirac delta function discussed in subsection 13.1.3. From (30.42)

and the fundamental property of the delta function (13.12), we see that

Pr(a<X≤b)=

∫b

a

f(x)dx,

=

∑n

i=1

pi

∫b

a

δ(x−xi)dx=

∑

i

pi,

where the final sum extends over those values ofifor whicha<xi≤b.

30.4.3 Sets of random variables

It is common in practice to consider two or more random variables simultane-

ously. For example, one might be interested in both the height and weight of

a person drawn at random from a population. In the general case, these vari-

ables may depend on one another and are described byjoint probability density

functions; these are discussed fully in section 30.11. We simply note here that if

we have (say) two random variablesXandYthen by analogy with the single-

variable case we define their joint probability density functionf(x, y)insucha

way that, ifXandYare discrete RVs,

Pr(X=xi,Y=yj)=f(xi,yj),

or, ifXandYare continuous RVs,

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