Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

56 Probability and Distributions

Example 1.7.6.LetXhave the pdf

f(x)=

{
4 x^30 <x< 1
0elsewhere.
Consider the random variableY =−logX. Here are the steps of the above algo-
rithm:

1. The support ofY=−logXis (0,∞).


2. Ify=−logxthenx=e−y.


3. dxdy=−e−y.


4. Thus the pdf ofYis:

fY(y)=fX

(

e−y

)∣

∣−e−y

∣

∣=4(e−y)^3 e−y=4e−^4 y.

1.7.3 Mixtures of Discrete and Continuous Type Distribu-

tions

We close this section by two examples of distributions that are not of the discrete
or the continuous type.

Example 1.7.7.Let a distribution function be given by

F(x)=

⎧

⎨

⎩

0 x< 0

x+1

2 0 ≤x<^1

11 ≤x.

Then, for instance,

P

(

− 3 <X≤

1

2

)

=F

(

1

2

)

−F(−3) =

3

4

−0=

3

4

and

P(X=0)=F(0)−F(0−)=

1

2

−0=

1

2

.

The graph ofF(x) is shown in Figure 1.7.3. We see thatF(x) is not always
continuous, nor is it a step function. Accordingly, the corresponding distribution is
neither of the continuous type nor of the discrete type. It may be described as a
mixtureof those types.

Distributions that are mixtures of the continuous and discrete type do, in fact,
occur frequently in practice. For illustration, in life testing, suppose we know that
the length of life, sayX, exceeds the numberb, but the exact value ofXis unknown.
This is calledcensoring. For instance, this can happen when a subject in a cancer
study simply disappears; the investigator knows that the subject has lived a certain
number of months, but the exact length of life is unknown. Or it might happen
when an investigator does not have enough time in an investigation to observe the
moments of deaths of all the animals, say rats, in some study. Censoring can also
occur in the insurance industry; in particular, consider a loss with a limited-pay
policy in which the top amount is exceeded but it is not known by how much.