Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

38 Probability and Distributions

As Exercise 1.5.11 shows,PX(D) is a probability onD. An example is helpful here.

Example 1.5.1 (First Roll in the Game of Craps). LetX be the sum of the
upfaces on a roll of a pair of fair 6-sided dice, each with the numbers 1 through 6
on it. The sample space isC={(i, j): 1≤i, j≤ 6 }. Because the dice are fair,
P[{(i, j)}]=1/36. The random variableXisX(i, j)=i+j.ThespaceofXis
D={ 2 ,..., 12 }. By enumeration, the pmf ofXis given by

Range value x 2 3 4 5 6 7 8 9 10 11 12

Probability pX(x) 361 362 363 364 365 366 365 364 363 362 361

To illustrate the computation of probabilities concerningX, supposeB 1 ={x:x=
7 , 11 }andB 2 ={x: x=2, 3 , 12 }. Then, using the values ofpX(x) given in the
table,

PX(B 1 )=

∑

x∈B 1

pX(x)=

6

36

+

2

36

=

8

36

PX(B 2 )=

∑

x∈B 2

pX(x)=

1

36

+

2

36

+

1

36

=

4

36

.

The second case is whenXis a continuous random variable. In this case,D
is an interval of real numbers. In practice, continuous random variables are often
measurements. For example, the weight of an adult is modeled by a continuous
random variable. Here we would not be interested in the probability that a person
weighs exactly 200 pounds, but we may be interested in the probability that a
person weighs over 200 pounds. Generally, for the continuous random variables,
the simple events of interest are intervals. We can usually determine a nonnegative
functionfX(x) such that for any interval of real numbers (a, b)∈D, the induced
probability distribution ofX,PX(·), is defined as

PX[(a, b)] =P[{c∈C:a<X(c)<b}]=

∫b

a

fX(x)dx; (1.5.2)

that is, the probability thatXfalls betweenaandbis the area under the curve
y∫=fX(x) betweenaandb. BesidesfX(x)≥0, we also require thatPX(D)=

DfX(x)dx= 1 (total area under the curve over the sample space ofXis 1). There
are some technical issues in defining events in general for the spaceD; however, it
can be shown thatPX(D) is a probability onD; see Exercise 1.5.11. The function
fXis formally defined as theprobability density function(pdf)ofXin Section
1.7. An example is in order.

Example 1.5.2. For an example of a continuous random variable, consider the
following simple experiment: choose a real number at random from the interval
(0,1). LetXbe the number chosen. In this case the space ofXisD=(0,1). It is
not obvious as it was in the last example what the induced probabilityPXis. But