Finding Cumulative distribution function of $f_x(X) = frac14 cdot |X|$Cumulative distribution function and expected valueFinding Cumulative Distribution Function given two independent pdfsFind continuous stochastic variable $X$ with PDF $f_X = frac1x^2$Cumulative distribution function of sub of two random variableFinding the probability density from cumulative distribution functioncumulative distribution function method transformationFinding density function given a cumulative distributionCumulative Distribution Function for a Certain rangeCummulative Distribution Function: Sum of two independent exp-distributed random variablesCumulative distribution function to probability density

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Finding Cumulative distribution function of $f_x(X) = frac14 cdot |X|$


Cumulative distribution function and expected valueFinding Cumulative Distribution Function given two independent pdfsFind continuous stochastic variable $X$ with PDF $f_X = frac1x^2$Cumulative distribution function of sub of two random variableFinding the probability density from cumulative distribution functioncumulative distribution function method transformationFinding density function given a cumulative distributionCumulative Distribution Function for a Certain rangeCummulative Distribution Function: Sum of two independent exp-distributed random variablesCumulative distribution function to probability density













0












$begingroup$



Let $f_X(x) = frac14cdot |X|$ a probability density function.



$-2<X<2$;



Find the Cumulative distribution function of $f_X(x)$.




What I thought was the answer is $F_X = frac12 + fracx8$. Since a CDF is a function which tells us the probability of a $P(X<a)$, given certain.



Drawing the PDF generats a graph of a symetric function, which gives us a probability of a least $frac12$.



However I found out it is $F_x = PleftXle xright=frac12-frac12cdot |x|cdot |fracx4|= frac12 - fracx^28$, for $-2 < x < 0$.



I can't understand why.










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    We have $P(Xle x) =int_-2^x frac14|t|, dt $ if $xin (-2,2)$. Did you try using this? This probability is only at least $1/2$ if $xge 0$. To compute this integral, you can try doing it by cases of $x < 0$ and $x ge 0$ (for $xin (-2,2)$).
    $endgroup$
    – Minus One-Twelfth
    2 days ago







  • 1




    $begingroup$
    Do you mean it is $frac12 + fracx^28$ for $0 < x < 2$?
    $endgroup$
    – Infiaria
    2 days ago










  • $begingroup$
    @Infiaria Yes, it is. Thank you.
    $endgroup$
    – Alan
    2 days ago










  • $begingroup$
    @MinusOne-Twelfth Got it to work, thanks.
    $endgroup$
    – Alan
    2 days ago















0












$begingroup$



Let $f_X(x) = frac14cdot |X|$ a probability density function.



$-2<X<2$;



Find the Cumulative distribution function of $f_X(x)$.




What I thought was the answer is $F_X = frac12 + fracx8$. Since a CDF is a function which tells us the probability of a $P(X<a)$, given certain.



Drawing the PDF generats a graph of a symetric function, which gives us a probability of a least $frac12$.



However I found out it is $F_x = PleftXle xright=frac12-frac12cdot |x|cdot |fracx4|= frac12 - fracx^28$, for $-2 < x < 0$.



I can't understand why.










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    We have $P(Xle x) =int_-2^x frac14|t|, dt $ if $xin (-2,2)$. Did you try using this? This probability is only at least $1/2$ if $xge 0$. To compute this integral, you can try doing it by cases of $x < 0$ and $x ge 0$ (for $xin (-2,2)$).
    $endgroup$
    – Minus One-Twelfth
    2 days ago







  • 1




    $begingroup$
    Do you mean it is $frac12 + fracx^28$ for $0 < x < 2$?
    $endgroup$
    – Infiaria
    2 days ago










  • $begingroup$
    @Infiaria Yes, it is. Thank you.
    $endgroup$
    – Alan
    2 days ago










  • $begingroup$
    @MinusOne-Twelfth Got it to work, thanks.
    $endgroup$
    – Alan
    2 days ago













0












0








0





$begingroup$



Let $f_X(x) = frac14cdot |X|$ a probability density function.



$-2<X<2$;



Find the Cumulative distribution function of $f_X(x)$.




What I thought was the answer is $F_X = frac12 + fracx8$. Since a CDF is a function which tells us the probability of a $P(X<a)$, given certain.



Drawing the PDF generats a graph of a symetric function, which gives us a probability of a least $frac12$.



However I found out it is $F_x = PleftXle xright=frac12-frac12cdot |x|cdot |fracx4|= frac12 - fracx^28$, for $-2 < x < 0$.



I can't understand why.










share|cite|improve this question











$endgroup$





Let $f_X(x) = frac14cdot |X|$ a probability density function.



$-2<X<2$;



Find the Cumulative distribution function of $f_X(x)$.




What I thought was the answer is $F_X = frac12 + fracx8$. Since a CDF is a function which tells us the probability of a $P(X<a)$, given certain.



Drawing the PDF generats a graph of a symetric function, which gives us a probability of a least $frac12$.



However I found out it is $F_x = PleftXle xright=frac12-frac12cdot |x|cdot |fracx4|= frac12 - fracx^28$, for $-2 < x < 0$.



I can't understand why.







probability probability-theory density-function






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago







Alan

















asked 2 days ago









AlanAlan

1,3991021




1,3991021







  • 1




    $begingroup$
    We have $P(Xle x) =int_-2^x frac14|t|, dt $ if $xin (-2,2)$. Did you try using this? This probability is only at least $1/2$ if $xge 0$. To compute this integral, you can try doing it by cases of $x < 0$ and $x ge 0$ (for $xin (-2,2)$).
    $endgroup$
    – Minus One-Twelfth
    2 days ago







  • 1




    $begingroup$
    Do you mean it is $frac12 + fracx^28$ for $0 < x < 2$?
    $endgroup$
    – Infiaria
    2 days ago










  • $begingroup$
    @Infiaria Yes, it is. Thank you.
    $endgroup$
    – Alan
    2 days ago










  • $begingroup$
    @MinusOne-Twelfth Got it to work, thanks.
    $endgroup$
    – Alan
    2 days ago












  • 1




    $begingroup$
    We have $P(Xle x) =int_-2^x frac14|t|, dt $ if $xin (-2,2)$. Did you try using this? This probability is only at least $1/2$ if $xge 0$. To compute this integral, you can try doing it by cases of $x < 0$ and $x ge 0$ (for $xin (-2,2)$).
    $endgroup$
    – Minus One-Twelfth
    2 days ago







  • 1




    $begingroup$
    Do you mean it is $frac12 + fracx^28$ for $0 < x < 2$?
    $endgroup$
    – Infiaria
    2 days ago










  • $begingroup$
    @Infiaria Yes, it is. Thank you.
    $endgroup$
    – Alan
    2 days ago










  • $begingroup$
    @MinusOne-Twelfth Got it to work, thanks.
    $endgroup$
    – Alan
    2 days ago







1




1




$begingroup$
We have $P(Xle x) =int_-2^x frac14|t|, dt $ if $xin (-2,2)$. Did you try using this? This probability is only at least $1/2$ if $xge 0$. To compute this integral, you can try doing it by cases of $x < 0$ and $x ge 0$ (for $xin (-2,2)$).
$endgroup$
– Minus One-Twelfth
2 days ago





$begingroup$
We have $P(Xle x) =int_-2^x frac14|t|, dt $ if $xin (-2,2)$. Did you try using this? This probability is only at least $1/2$ if $xge 0$. To compute this integral, you can try doing it by cases of $x < 0$ and $x ge 0$ (for $xin (-2,2)$).
$endgroup$
– Minus One-Twelfth
2 days ago





1




1




$begingroup$
Do you mean it is $frac12 + fracx^28$ for $0 < x < 2$?
$endgroup$
– Infiaria
2 days ago




$begingroup$
Do you mean it is $frac12 + fracx^28$ for $0 < x < 2$?
$endgroup$
– Infiaria
2 days ago












$begingroup$
@Infiaria Yes, it is. Thank you.
$endgroup$
– Alan
2 days ago




$begingroup$
@Infiaria Yes, it is. Thank you.
$endgroup$
– Alan
2 days ago












$begingroup$
@MinusOne-Twelfth Got it to work, thanks.
$endgroup$
– Alan
2 days ago




$begingroup$
@MinusOne-Twelfth Got it to work, thanks.
$endgroup$
– Alan
2 days ago










1 Answer
1






active

oldest

votes


















1












$begingroup$

With the help above, I made it -



I will show the answer for the interval $-2 < x < 0$. The other one is symetric.



$$int_-2^x frac14cdot |t| dt = int_-2^x frac14cdot (-t) dt $$ (since t is negative).
$$=frac14cdot int_-2^x (t) dt =- frac14cdot fract^22|_-2^x=-frac14cdot left( fracx^22 - 2 right) =frac12 - fracx^28 $$.



So at the end we get -



$$F_x = left{ beginarray*20l0&X < - 2\beginarraylfrac12 - fracx^28\frac12 + fracx^28\1endarray&beginarrayl - 2 < x < 0\0 < x < 2\x > 2endarrayendarray right.
% MathType!MTEF!2!1!+-
% feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj
% MCPbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B
% TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8
% qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9
% q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake
% aaqaaaaaaaaaWdbiaadAeapaWaaSbaaSqaa8qacaWG4baapaqabaGc
% peGaeyypa0Zaaiqaa8aabaqbaeaabiGaaaqaaiaaicdaaeaacaWGyb
% GaeyipaWJaeyOeI0IaaGOmaaabaeqabaWaaSaaaeaacaaIXaaabaGa
% aGOmaaaacqGHsisldaWcaaqaaiaadIhadaahaaWcbeqaaiaaikdaaa
% aakeaacaaI4aaaaaqaamaalaaabaGaaGymaaqaaiaaikdaaaGaey4k
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% qaaiaadIhacqGH+aGpcaaIYaaaaaaapeGaay5Eaaaaaa!5984!
$$






share|cite|improve this answer









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    1 Answer
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    oldest

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    1 Answer
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    active

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    active

    oldest

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    1












    $begingroup$

    With the help above, I made it -



    I will show the answer for the interval $-2 < x < 0$. The other one is symetric.



    $$int_-2^x frac14cdot |t| dt = int_-2^x frac14cdot (-t) dt $$ (since t is negative).
    $$=frac14cdot int_-2^x (t) dt =- frac14cdot fract^22|_-2^x=-frac14cdot left( fracx^22 - 2 right) =frac12 - fracx^28 $$.



    So at the end we get -



    $$F_x = left{ beginarray*20l0&X < - 2\beginarraylfrac12 - fracx^28\frac12 + fracx^28\1endarray&beginarrayl - 2 < x < 0\0 < x < 2\x > 2endarrayendarray right.
    % MathType!MTEF!2!1!+-
    % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj
    % MCPbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B
    % TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8
    % qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9
    % q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake
    % aaqaaaaaaaaaWdbiaadAeapaWaaSbaaSqaa8qacaWG4baapaqabaGc
    % peGaeyypa0Zaaiqaa8aabaqbaeaabiGaaaqaaiaaicdaaeaacaWGyb
    % GaeyipaWJaeyOeI0IaaGOmaaabaeqabaWaaSaaaeaacaaIXaaabaGa
    % aGOmaaaacqGHsisldaWcaaqaaiaadIhadaahaaWcbeqaaiaaikdaaa
    % aakeaacaaI4aaaaaqaamaalaaabaGaaGymaaqaaiaaikdaaaGaey4k
    % aSYaaSaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGioaa
    % aaaeaacaaIXaaaaqaabeqaaiabgkHiTiaaikdacqGH8aapcaWG4bGa
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    % qaaiaadIhacqGH+aGpcaaIYaaaaaaapeGaay5Eaaaaaa!5984!
    $$






    share|cite|improve this answer









    $endgroup$

















      1












      $begingroup$

      With the help above, I made it -



      I will show the answer for the interval $-2 < x < 0$. The other one is symetric.



      $$int_-2^x frac14cdot |t| dt = int_-2^x frac14cdot (-t) dt $$ (since t is negative).
      $$=frac14cdot int_-2^x (t) dt =- frac14cdot fract^22|_-2^x=-frac14cdot left( fracx^22 - 2 right) =frac12 - fracx^28 $$.



      So at the end we get -



      $$F_x = left{ beginarray*20l0&X < - 2\beginarraylfrac12 - fracx^28\frac12 + fracx^28\1endarray&beginarrayl - 2 < x < 0\0 < x < 2\x > 2endarrayendarray right.
      % MathType!MTEF!2!1!+-
      % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj
      % MCPbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B
      % TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8
      % qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9
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      % aaqaaaaaaaaaWdbiaadAeapaWaaSbaaSqaa8qacaWG4baapaqabaGc
      % peGaeyypa0Zaaiqaa8aabaqbaeaabiGaaaqaaiaaicdaaeaacaWGyb
      % GaeyipaWJaeyOeI0IaaGOmaaabaeqabaWaaSaaaeaacaaIXaaabaGa
      % aGOmaaaacqGHsisldaWcaaqaaiaadIhadaahaaWcbeqaaiaaikdaaa
      % aakeaacaaI4aaaaaqaamaalaaabaGaaGymaaqaaiaaikdaaaGaey4k
      % aSYaaSaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGioaa
      % aaaeaacaaIXaaaaqaabeqaaiabgkHiTiaaikdacqGH8aapcaWG4bGa
      % eyipaWJaaGimaaqaaiaaicdacqGH8aapcaWG4bGaeyipaWJaaGOmaa
      % qaaiaadIhacqGH+aGpcaaIYaaaaaaapeGaay5Eaaaaaa!5984!
      $$






      share|cite|improve this answer









      $endgroup$















        1












        1








        1





        $begingroup$

        With the help above, I made it -



        I will show the answer for the interval $-2 < x < 0$. The other one is symetric.



        $$int_-2^x frac14cdot |t| dt = int_-2^x frac14cdot (-t) dt $$ (since t is negative).
        $$=frac14cdot int_-2^x (t) dt =- frac14cdot fract^22|_-2^x=-frac14cdot left( fracx^22 - 2 right) =frac12 - fracx^28 $$.



        So at the end we get -



        $$F_x = left{ beginarray*20l0&X < - 2\beginarraylfrac12 - fracx^28\frac12 + fracx^28\1endarray&beginarrayl - 2 < x < 0\0 < x < 2\x > 2endarrayendarray right.
        % MathType!MTEF!2!1!+-
        % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj
        % MCPbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B
        % TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8
        % qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9
        % q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake
        % aaqaaaaaaaaaWdbiaadAeapaWaaSbaaSqaa8qacaWG4baapaqabaGc
        % peGaeyypa0Zaaiqaa8aabaqbaeaabiGaaaqaaiaaicdaaeaacaWGyb
        % GaeyipaWJaeyOeI0IaaGOmaaabaeqabaWaaSaaaeaacaaIXaaabaGa
        % aGOmaaaacqGHsisldaWcaaqaaiaadIhadaahaaWcbeqaaiaaikdaaa
        % aakeaacaaI4aaaaaqaamaalaaabaGaaGymaaqaaiaaikdaaaGaey4k
        % aSYaaSaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGioaa
        % aaaeaacaaIXaaaaqaabeqaaiabgkHiTiaaikdacqGH8aapcaWG4bGa
        % eyipaWJaaGimaaqaaiaaicdacqGH8aapcaWG4bGaeyipaWJaaGOmaa
        % qaaiaadIhacqGH+aGpcaaIYaaaaaaapeGaay5Eaaaaaa!5984!
        $$






        share|cite|improve this answer









        $endgroup$



        With the help above, I made it -



        I will show the answer for the interval $-2 < x < 0$. The other one is symetric.



        $$int_-2^x frac14cdot |t| dt = int_-2^x frac14cdot (-t) dt $$ (since t is negative).
        $$=frac14cdot int_-2^x (t) dt =- frac14cdot fract^22|_-2^x=-frac14cdot left( fracx^22 - 2 right) =frac12 - fracx^28 $$.



        So at the end we get -



        $$F_x = left{ beginarray*20l0&X < - 2\beginarraylfrac12 - fracx^28\frac12 + fracx^28\1endarray&beginarrayl - 2 < x < 0\0 < x < 2\x > 2endarrayendarray right.
        % MathType!MTEF!2!1!+-
        % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj
        % MCPbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B
        % TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8
        % qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9
        % q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake
        % aaqaaaaaaaaaWdbiaadAeapaWaaSbaaSqaa8qacaWG4baapaqabaGc
        % peGaeyypa0Zaaiqaa8aabaqbaeaabiGaaaqaaiaaicdaaeaacaWGyb
        % GaeyipaWJaeyOeI0IaaGOmaaabaeqabaWaaSaaaeaacaaIXaaabaGa
        % aGOmaaaacqGHsisldaWcaaqaaiaadIhadaahaaWcbeqaaiaaikdaaa
        % aakeaacaaI4aaaaaqaamaalaaabaGaaGymaaqaaiaaikdaaaGaey4k
        % aSYaaSaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGioaa
        % aaaeaacaaIXaaaaqaabeqaaiabgkHiTiaaikdacqGH8aapcaWG4bGa
        % eyipaWJaaGimaaqaaiaaicdacqGH8aapcaWG4bGaeyipaWJaaGOmaa
        % qaaiaadIhacqGH+aGpcaaIYaaaaaaapeGaay5Eaaaaaa!5984!
        $$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 2 days ago









        AlanAlan

        1,3991021




        1,3991021



























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