Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

48 Probability and Distributions

for there are no negative values ofxinDX={x:x=0, 1 , 2 , 3 }. That is, we have

the single-valued inverse functionx=g−^1 (y)=

√

y(not−

√

y), and so

pY(y)=pX(

√

y)=

3!

(

√

y)!(3−

√

y)!

(

2

3

)√y(

1

3

) 3 −√y

,y=0, 1 , 4 , 9.

The second case is where the transformation,g(x), is not one-to-one. Instead of
developing an overall rule, for most applications involving discrete random variables
the pmf ofYcan be obtained in a straightforward manner. We offer two examples
as illustrations.
Consider the geometric random variable in Example 1.6.3. Suppose we are
playing a game against the “house”(say, a gambling casino). If the first head appears
on an odd number of flips, we pay the house one dollar, while if it appears on an
even number of flips, we win one dollar from the house. LetYdenote our net gain.
Then the space ofY is{− 1 , 1 }. In Example 1.6.1, we showed that the probability
thatXis odd is^23. Hence, the distribution ofY is given bypY(−1) = 2/3and
pY(1) = 1/3.
As a second illustration, letZ=(X−2)^2 ,whereXis the geometric random
variable of Example 1.6.1. Then the space ofZisDZ ={ 0 , 1 , 4 , 9 , 16 ,...}.Note
thatZ=0ifandonlyifX=2;Z=1ifandonlyifX=1orX= 3; while for the
other values of the space there is a one-to-one correspondence given byx=

√
z+2,
forz∈{ 4 , 9 , 16 ,...}. Hence, the pmf ofZis

pZ(z)=

⎧

⎨

⎩

pX(2) =^14 forz=0

pX(1) +pX(3) =^58 forz=1

pX(

√

z+2)=^14

( 1

2

)√z

forz=4, 9 , 16 ,....

(1.6.6)

For verification, the reader is asked to show in Exercise 1.6.11 that the pmf ofZ

sums to 1 over its space.

EXERCISES

1.6.1.LetXequal the number of heads in four independent flips of a coin. Using

certain assumptions, determine the pmf ofXand compute the probability thatX

is equal to an odd number.

1.6.2.Let a bowl contain 10 chips of the same size and shape. One and only one

of these chips is red. Continue to draw chips from the bowl, one at a time and at

random and without replacement, until the red chip is drawn.

(a)Find the pmf ofX, the number of trials needed to draw the red chip.

(b)ComputeP(X≤4).

1.6.3.Cast a die a number of independent times until a six appears on the up side

of the die.

(a)Find the pmfp(x)ofX, the number of casts needed to obtain that first six.