Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

46 Probability and Distributions

to obtain the first head. Hence,X(TTHTHHT···) = 3. Clearly, the space ofXis
D={ 1 , 2 , 3 , 4 ,...}.WeseethatX= 1 when the sequence begins with an H and
thusP(X=1)=^12. Likewise,X= 2 when the sequence begins with TH, which
has probabilityP(X=2)=(^12 )(^12 )=^14 from the independence. More generally,
ifX =x,wherex=1, 2 , 3 , 4 ,...,there must be a string ofx−1 tails followed
by a head; that is, TT···TH, where there arex−1 tails in TT···T. Thus, from
independence, we have a geometric sequence of probabilities, namely,

P(X=x)=

(

1

2

)x− 1 (

1

2

)

=

(

1

2

)x

,x=1, 2 , 3 ,..., (1.6.1)

the space of which is countable. An interesting event is that the first head appears

on an odd number of flips; i.e.,X∈{ 1 , 3 , 5 ,...}. The probability of this event is

P[X∈{ 1 , 3 , 5 ,...}]=

∑∞

x=1

(

1

2

) 2 x− 1

=

1

2

∑∞

x=1

(

1

4

)x− 1

=

1 / 2

1 −(1/4)

=

2

3

.

As the last example suggests, probabilities concerning a discrete random vari-
able can be obtained in terms of the probabilitiesP(X =x), forx∈D.These
probabilities determine an important function, which we define as

Definition 1.6.2(Probability Mass Function (pmf)).LetXbe a discrete random
variable with spaceD.Theprobability mass function(pmf) ofXis given by

pX(x)=P[X=x], forx∈D. (1.6.2)

Note that pmfs satisfy the following two properties:

(i) 0≤pX(x)≤ 1 ,x∈D, and (ii)

∑

x∈DpX(x)=1. (1.6.3)

In a more advanced class it can be shown that if a function satisfies properties (i)
and (ii) for a discrete setD, then this function uniquely determines the distribution
of a random variable.
LetXbe a discrete random variable with spaceD. As Theorem 1.5.3 shows,
discontinuities ofFX(x) define a mass; that is, ifxis a point of discontinuity ofFX,
thenP(X=x)>0. We now make a distinction between the space of a discrete
random variable and these points of positive probability. We define thesupportof
a discrete random variableXto be the points in the space ofXwhich have positive
probability. We often useSto denote the support ofX.NotethatS⊂D, but it
may be thatS=D.
Also, we can use Theorem 1.5.3 to obtain a relationship between the pmf and
cdf of a discrete random variable. Ifx∈S,thenpX(x) is equal to the size of the
discontinuity ofFXatx.Ifx
∈SthenP[X=x] = 0 and, hence,FXis continuous
at thisx.

Example 1.6.2.A lot, consisting of 100 fuses, is inspected by the following proce-
dure. Five of these fuses are chosen at random and tested; if all five “blow” at the