Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

1.6. Discrete Random Variables 47

correct amperage, the lot is accepted. If, in fact, there are 20 defective fuses in the

lot, the probability of accepting the lot is, under appropriate assumptions,

( 80

5

)

( 100

5

)=0. 31931.

More generally, let the random variableXbe the number of defective fuses among
the five that are inspected. The pmf ofXis given by

pX(x)=

{ ( 20

x)(

80

5 −x)

(^1005 )

forx=0, 1 , 2 , 3 , 4 , 5

0elsewhere.

(1.6.4)

Clearly, the space ofXisD={ 0 , 1 , 2 , 3 , 4 , 5 }, which is also its support. This is an

example of a random variable of the discrete type whose distribution is an illustra-

tion of ahypergeometric distribution, which is formally defined in Chapter 3.

Based on the above discussion, it is easy to graph the cdf ofX; see Exercise 1.6.5.

1.6.1 Transformations

A problem often encountered in statistics is the following. We have a random
variableXand we know its distribution. We are interested, though, in a random
variableY which is sometransformationofX,say,Y =g(X). In particular,
we want to determine the distribution ofY. AssumeXis discrete with spaceDX.
Then the space ofYisDY={g(x):x∈DX}. We consider two cases.
In the first case,gis one-to-one. Then, clearly, the pmf ofY is obtained as

pY(y)=P[Y=y]=P[g(X)=y]=P[X=g−^1 (y)] =pX(g−^1 (y)). (1.6.5)

Example 1.6.3.Consider the random variableXof Example 1.6.1. Recall thatX
was the flip number on which the first head appeared. LetYbe the number of flips
before the first head. ThenY=X−1. In this case, the functiongisg(x)=x−1,
whose inverse is given byg−^1 (y)=y+1. The space ofYisDY={ 0 , 1 , 2 ,...}.The
pmf ofXis given by (1.6.1); hence, based on expression (1.6.5), the pmf ofY is

pY(y)=pX(y+1)=

(

1

2

)y+1

, fory=0, 1 , 2 ,....

Example 1.6.4.LetXhave the pmf

pX(x)=

{

3!

x!(3−x)!

( 2

3

)x( 1

3

) 3 −x

x=0, 1 , 2 , 3

0elsewhere.

We seek the pmfpY(y) of the random variableY =X^2. The transformation
y=g(x)=x^2 mapsDX={x:x=0, 1 , 2 , 3 }ontoDY ={y:y=0, 1 , 4 , 9 }.In
general,y=x^2 does not define a one-to-one transformation; here, however, it does,