Transcription of The Cumulative Distribution Function for a Random Variable
1 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 under its graph. Moreareasprecisely, the probability that a value of is between and .\+, T + \ , 0 B .B'+,For example, T " \ $ 0 B .B'"$ T $ \ T $ \ _ 0 B .B'$_ T \ " T _ \ " 0 B .B' _ " i) Since probabilities are always between and , it must be that !"0 B ! (so that can never give a negative probability ), and'+,0 B.
2 B ii) Since a certain event has probability ," total area under the graph of T _ \ _ " 0 B .B 0 B ' __The properties i) and ii) are necessary for a Function to be the pdf for some random0 B Variable \ We can also use property ii) in computations: since ''' _ _$_$_0 B .B 0 B 0 B .B " T \ $ 0 B .B " 0 B .B " T \ $ '' _$$_The pdf is discussed in the is another Function , the (cdf) which records thecumulative Distribution functionsame probabilities associated with , but in a different way.
3 The cdf is defined by\J B .J B T \ B J B Bgives the accumulated probability up to . We can see immediately how thepdf and cdf are related: (since is used as a Variable in theJ B T \ B 0 > .>B' _B upper limit of integration, we use some other Variable , say , in the integrand)>Notice that (since it's a probability), and thatJ B ! a) andlimlimB _B _ _ _B_J B 0 > .> 0 > .> "'' b) , and thatlimlimB _B _ _ _B _J B 0 >.
4 > 0 > .> !'' c) (by the Fundamental Theorem of Calculus)J B 0 B wItem c) states the connection between the cdf and pdf in another way: (the particular antiderivativethe cdfis an antiderivative of the pdfJ B 0 B where the constant of integration is chosen to make the limit in a) true)and thereforeT + \ , 0 B .B J B l J , J + T \ , T \ + '+,+,_____Example: Suppose has an exponential density Function . As discussed in class,\ (where 0 B - !
5 B !-/B ! -B".If , , soB !0 > .> 0 > .> -/ .> / l " /''' _!!BBB -> -> B -B! J B !B !" /B ! -BIf has mean, say, then .\ $- .""$.If we want to know , we can either computeT \ % '' _ _%%"$ " $ B0 B .B /.B ! ($'%!$J B , or (now that we have the formula for we can simply computeJ $ " / " / ! ($'%!$ " $ % % $(The graphs of and are shown on the last page before exercises. In the figure,0 B J B notice the values ofand limlimB _B _J B J B _____Example: If is a normal Random Variable with mean and standard deviation\ !))))
6 5 " 0 B /J B /.>then its pdf is , and its cdf .""## B # > # _B 11##'Because there is no elementary antiderivative for , its not possible to find an/ > ## elementary formula for . However, for any , the value of canJ B B/ .>"# _B > # 1'#be estimated, so that a graph of can be drawn. (J B See figure on the last page beforeexercises.) Example: More generally, probability calculations involving a normal Random Variable \ are computationally difficult because again there's no elementary formula for thecumulative Distribution Function that is, an antiderivative for the probabilityJ B den ity Function = 0 B /"# B # ##Therefore it's not possible to find an exact value for T + \ , /.
7 B J , J + '+,"# B # ##Suppose is a normal Random Variable with mean and standard deviation\ " *.5 " (T $ \ # . If we want to find , we need to estimate " " ( # B " * # " ( 1'32/.B J # J $ ##This can be done with Simpson's Rule. However, such calculations are so important thatthe TI83-Plus Calculator has a built in way to make the estimate: Punch keys normalcdfChoose item 2 on the menu: On the screen you see normalcdf Fill in normalcdf $ # " * " ( and the TI-83 gives the approximate value of the integral above: 480!))))
8 &#"The general syntax for thecommand is normalcdf (lowerlimit,upperlimit,).5 If you enter normalcdfonly lowerlimit,upperlimit then the TI-83 assumes .5 ! "as the default valuesNote that using the values for example given above:.5 normalcdfT \ # $ ' " * " ( !)
9 ')#(.5 .5 normalcdfT # \ # " & & $ " * " ( ! *&%&.5 .5 normalcdfT $ \ $ $ # ( " * " ( ! **($.5 .5In fact (as may have been mentioned in class) these probabilities come out the same forany Random Variable , no matter what the values of and : for example, that normal Random Variable takes on a value between one standardany deviation of its mean is Exercises:1.))))
10 A certain uniform Random Variable has pdf otherwise.\0 B " & # B (! 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 " " B 1# a) What is ?T \ " b) Write the formula for its cdf J B c) Write a formula using that gives the answer to part a). Check that itJ B agrees with your numerical answer in a).
