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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