Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

50 Probability and Distributions

for some functionfX(t). The functionfX(t) is called aprobability density func-
tion(pdf) ofX.IffX(x) is also continuous, then the Fundamental Theorem of
Calculus implies that
d
dx

FX(x)=fX(x). (1.7.2)

Thesupportof a continuous random variableXconsists of all pointsxsuch
thatfX(x)>0. As in the discrete case, we often denote the support ofXbyS.
IfXis a continuous random variable, then probabilities can be obtained by
integration; i.e.,

P(a<X≤b)=FX(b)−FX(a)=

∫b

a

fX(t)dt.

Also, for continuous random variables,

P(a<X≤b)=P(a≤X≤b)=P(a≤X<b)=P(a<X<b).

From the definition (1.7.2), note that pdfs satisfy the two properties

(i)fX(x)≥0 and (ii)

∫∞

−∞fX(t)dt=1. (1.7.3)

The second property, of course, follows fromFX(∞) = 1. In an advanced course in
probability, it is shown that if a function satisfies the above two properties, then it
is a pdf for a continuous random variable; see, for example, Tucker (1967).
Recall in Example 1.5.2 the simple experiment where a number was chosen
at random from the interval (0,1). The number chosen,X,isanexampleofa
continuous random variable. Recall that the cdf ofXisFX(x)=x,for0<x<1.
Hence, the pdf ofXis given by

fX(x)=

{

10 <x< 1

0elsewhere.

(1.7.4)

Any continuous or discrete random variableXwhose pdf or pmf is constant on
the support ofXis said to have auniformdistribution; see Chapter 3 for a more
formal definition.

Example 1.7.1(Point Chosen at Random Within the Unit Circle). Suppose we
select a point at random in the interior of a circle of radius 1. Let X be the
distance of the selected point from the origin. The sample space for the experiment
isC={(w, y):w^2 +y^2 < 1 }. Because the point is chosen at random, it seems
that subsets ofCwhich have equal area are equilikely. Hence, the probability of the
selected point lying in a setA⊂Cis proportional to the area ofA; i.e.,

P(A)=

area ofA

π

.

For 0<x<1, the event{X≤x}is equivalent to the point lying in a circle of
radiusx. By this probability rule,P(X≤x)=πx^2 /π=x^2 ; hence, the cdf ofXis

FX(x)=

⎧

⎨

⎩

0 x< 0

x^20 ≤x< 1

11 ≤x.

(1.7.5)