The Cumulative Distribution Function for a Random Variable
The Cumulative Distribution Function for a Random Variable \
Each continuous random variable \ has an associated probability density function (pdf)
0 ?B?. It ¡°records¡± the probabilities associated with \ as areas under its graph. More
precisely,
¡°the probability that a value of \ is between + and ,¡± ? T ?+ ? \ ? ,? ? '+ 0 ?B? .B.
For example,
$
T ?" ? \ ? $?
? '" 0 ?B? .B
_
T ?$ ? \? ? T ?$ ? \ ? _? ? '$ 0 ?B? .B
?"
T ?\ ? ? "? ? T ? ? _ ? \ ? ? "? ? '?_ 0 ?B? .B
,
i) Since probabilities are always between ! and ", it must be that 0 ?B? !
,
(so that '+ 0 ?B? .B can never give a ¡°negative probability¡±), and
ii) Since a ¡°certain¡± event has probability ",
_
T ? ? _ ? \ ? _? ? " ? '?_ 0 ?B? .B ? total area under the graph of 0 ?B?
The properties i) and ii) are necessary for a function 0 ?B? to be the pdf for some random
variable \?
We can also use property ii) in computations: since
_
$
_
'?_
0 ?B? .B ? '?_ 0 ?B? ? '$ 0 ?B? .B ? "
$
_
T ?\ ? $? ? '?_ 0 ?B? .B ? " ? '$ 0 ?B? .B ? " ? T ?\
$?
The pdf is discussed in the textbook.
There is another function, the cumulative distribution function (cdf) which records the
same probabilities associated with \ , but in a different way. The cdf J ?B? is defined by
J ?B? ? T ?\ ? B?.
J ?B? gives the ¡°accumulated¡± probability ¡°up to B.¡± We can see immediately how the
pdf and cdf are related:
J ?B? ? T ?\ ? B? ? '?_ 0 ?>? .> (since ¡°B¡± is used as a variable in the
upper limit of integration, we use some
other variable, say ¡°>¡±, in the integrand)
B
Notice that J ?B?
! (since it's a probability), and that
a) lim J ?B? ? lim '?_ 0 ?>? .> ? '?_ 0 ?>? .> ? " and
B?_
B?_
B
?_
b) lim J ?B? ? lim ' 0 ?>? .> ? ' 0 ?>? .> ? !, and that
B
B??_
B??_ ?_
_
?_
c) J w ?B? ? 0 ?B? (by the Fundamental Theorem of Calculus)
Item c) states the connection between the cdf and pdf in another way:
the cdf J ?B? is an antiderivative of the pdf 0 ?B? (the particular antiderivative
where the constant of integration is chosen to make the limit in a) true)
and therefore
T ?+ ? \ ? ,? ? '+ 0 ?B? .B ? J ?B?l,+ ? J ?,? ? J ?+? ? T ?\ ? ,? ? T ?\ ? +?
,
________________________________________________________________________
Example: Suppose \ has an exponential density function. As discussed in class,
0 ?B? ? ?
If B
!
-/?-B
B?!
(where - ? ." ?
B !
!, '?_ 0 ?>? .> ? '! 0 ?>? .> ? '! -/?-> .> ? ? /?-> lB! ? " ? /?-B , so
B
B
J ?B? ? ?
B
!
" ? /?-B
B?!
B !
If \ has mean . ? $, say, then - ?
"
.
? "$ .
If we want to know T ?\ ? %?, we can either compute
%
%
'?_
0 ?B? .B ? '?_ " /??"?$ ?B .B ? !?($'%!$, or (now that we have the formula for J ?B?
$
we can simply compute J ?$? ? " ? /??"?$???% ? " ? /?%?$ ? !?($'%!$?
(The graphs of 0 ?B? and J ?B? are shown on the last page before exercises. In the figure,
notice the values of lim J ?B? and lim J ?B? ??
B?_
B??_
________________________________________________________________________
Example: If \ is a normal random variable with mean . ? ! and standard deviation
#
#
B
5 ? "? then its pdf is 0 ?B? ? ?"#1 /?B ?# , and its cdf J ?B? ? ?"#1 '?_ /?> ?# .>.
#
Because there is no ¡°elementary¡± antiderivative for /?> ?# , its not possible to find an
#
B
¡°elementary¡± formula for J ?B?. However, for any B, the value of ?"#1 '?_ /?> ?# .> can
be estimated, so that a graph of J ?B? can be drawn. (See figure on the last page before
exercises.)
Example: More generally, probability calculations involving a normal random variable
\ are computationally difficult because again there's no elementary formula for the
cumulative distribution function J ?B? ? that is, an antiderivative for the probability
den=ity function ?
0 ?B? ?
"
5 ?#1
#
/??B?.? ?#5
#
Therefore it's not possible to find an exact value for
T ?+ ? \ ? ,? ? '+
,
"
5 ?#1
#
#
/??B?.? ?#5 .B ? J ?,? ? J ?+?
Suppose \ is a normal random variable with mean . ? "?* and standard deviation
5 ? "?(. If we want to find T ? ? $ ? \ ? #?, we need to estimate
"
?"?(??#1
'?2 /??B?"?*?# ?#?"?(?# .B ? J ?#? ? J ? ? $??
3
This can be done with Simpson's Rule. However, such calculations are so important that
the TI83-Plus Calculator has a built in way to make the estimate:
Punch keys 28. HMWX V
Choose item 2 on the menu: normalcdf
On the screen you see
normalcdf ?
Fill in
normalcdf ? ? $? #? "?*? "?(?
and the TI-83 gives the approximate value of the integral above: !?"480
The general syntax for the
command is
If you enter only
then the TI-83 assumes . ? !? 5 ? "
as the default values
normalcdf (lowerlimit,upperlimit,.? 5 )
normalcdf ?lowerlimit,upperlimit?
Note that using the values for .? 5 example given above:
T ?. ? 5 ? \ ? . ? 5 ?
T ? . ? #5 ? \ ? . ? # 5 ?
T ? . ? $5 ? \ ? . ? $ 5 ?
? normalcdf ??#? $?'? "?*? "?(? ? !?')#(
? normalcdf ? ? "?&? &?$? "?*? "?(? ? !?*&%&
? normalcdf ? ? $?#? (? "?*? "?(? ? !?**($
In fact (as may have been mentioned in class) these probabilities come out the same for
any normal random variable, no matter what the values of . and 5 : for example, the
probability that any normal random variable takes on a value between ? one standard
deviation of its mean is ? 0.6827?
Exercises:
1. A certain ¡°uniform¡± random variable \ has pdf 0 ?B? ? ?
"?& # ? B ? (
!
otherwise.
a) What is T ?! ? \ ? $??
b) Write the formula for its cdf J ?B?
c) What is J ?$? ? J ?!? ?
2. A certain kind of random variable as density function 0 ?B? ?
a) What is T ?\
"
1 ?" ? B# ? .
? "??
b) Write the formula for its cdf J ?B?
c) Write a formula using J ?B? that gives the answer to part a). Check that it
agrees with your numerical answer in a).
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