What does the value of a PDF mean?












4












$begingroup$


I understand that the integral of a PDF provides tangible value --i.e., the integral of a PDF allows one to see the probability of a value or less than that value, under a particular distribution, occurring. But, what does the value of just the output of the PDF provide? In other words, what does the PDF of the standard normal distribution at x=0.5 mean?






probability







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  • 2




    $begingroup$
    The PDF is the rate of change of the CDF. No more, no less.
    $endgroup$
    – JimB
    2 hours ago






  • 1




    $begingroup$
    Possible duplicate of What does the value of a probability density function (PDF) at some x indicate?
    $endgroup$
    – Victoria M
    2 hours ago
















4












$begingroup$


I understand that the integral of a PDF provides tangible value --i.e., the integral of a PDF allows one to see the probability of a value or less than that value, under a particular distribution, occurring. But, what does the value of just the output of the PDF provide? In other words, what does the PDF of the standard normal distribution at x=0.5 mean?






probability







share|cite|improve this question







New contributor




Lea England is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$








  • 2




    $begingroup$
    The PDF is the rate of change of the CDF. No more, no less.
    $endgroup$
    – JimB
    2 hours ago






  • 1




    $begingroup$
    Possible duplicate of What does the value of a probability density function (PDF) at some x indicate?
    $endgroup$
    – Victoria M
    2 hours ago














4












4








4





$begingroup$


I understand that the integral of a PDF provides tangible value --i.e., the integral of a PDF allows one to see the probability of a value or less than that value, under a particular distribution, occurring. But, what does the value of just the output of the PDF provide? In other words, what does the PDF of the standard normal distribution at x=0.5 mean?






probability







share|cite|improve this question







New contributor




Lea England is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




I understand that the integral of a PDF provides tangible value --i.e., the integral of a PDF allows one to see the probability of a value or less than that value, under a particular distribution, occurring. But, what does the value of just the output of the PDF provide? In other words, what does the PDF of the standard normal distribution at x=0.5 mean?






probability




probability






share|cite|improve this question







New contributor




Lea England is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question







New contributor




Lea England is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question






New contributor




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Check out our Code of Conduct.









asked 2 hours ago









Lea EnglandLea England

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211




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New contributor





Lea England is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Lea England is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








  • 2




    $begingroup$
    The PDF is the rate of change of the CDF. No more, no less.
    $endgroup$
    – JimB
    2 hours ago






  • 1




    $begingroup$
    Possible duplicate of What does the value of a probability density function (PDF) at some x indicate?
    $endgroup$
    – Victoria M
    2 hours ago














  • 2




    $begingroup$
    The PDF is the rate of change of the CDF. No more, no less.
    $endgroup$
    – JimB
    2 hours ago






  • 1




    $begingroup$
    Possible duplicate of What does the value of a probability density function (PDF) at some x indicate?
    $endgroup$
    – Victoria M
    2 hours ago








2




2




$begingroup$
The PDF is the rate of change of the CDF. No more, no less.
$endgroup$
– JimB
2 hours ago




$begingroup$
The PDF is the rate of change of the CDF. No more, no less.
$endgroup$
– JimB
2 hours ago




1




1




$begingroup$
Possible duplicate of What does the value of a probability density function (PDF) at some x indicate?
$endgroup$
– Victoria M
2 hours ago




$begingroup$
Possible duplicate of What does the value of a probability density function (PDF) at some x indicate?
$endgroup$
– Victoria M
2 hours ago










3 Answers
3






active

oldest

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3












$begingroup$

I would first say that you should try to be OK with it not having a super down-to-earth meaning. It can sometimes help to let yourself be abstract and use the mathematical tools you have to get somewhere you might not have gotten by purely reasoning through with statements you might use to describe other things in your daily life. On the other hand, it is often helpful to have somewhat concrete ways of thinking about things that are abstract, so it's good that you're investigating this.



But as the PDF is the derivative of the CDF, the value of PDF $p(x)$ at $x$ is the (slope of the) best linear approximation of the CDF near $x$, i.e., that if $epsilon$ is super small, then if you want to approximate the probability that an event between $x$ and $x+epsilon$ happens linearly in $epsilon$, your best bet is to say $epsilon cdot p(x)$.






share|cite|improve this answer










New contributor




csprun is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$





















    1












    $begingroup$

    The value of the PDF is sort of like the "weighting" that each event gets for the CDF.






    share|cite|improve this answer








    New contributor




    Jacob Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






    $endgroup$





















      1












      $begingroup$


      Definition (Gut, 2005): A distribution function F is




      • discrete iff for some countable set of numbers {$x_j$} and point masses {$p_j$}, $$F(x) = sum_{x_jleq x} p_j, quad text{for all x} in Bbb R.$$ The function p is called the probability function.

      • continuous iff it is continuous for all x.

      • absolutely continuous iff there exists a non-negative, Lebesgue integrable function f, such that $$F(b) - F(a) = int_a^b f(x)text{ dx} quad {for space all space x < b.}$$ The function f is called the density of F.

      • singular iff F $neq$ 0, F' exists and equals 0 a.e.




      By the definition, evaluating a probability distribution function F is simply an integration of the density of said distribution. As mentioned, in the case for a $<$ b, this gives us the probability for a given range of values.



      However, if evaluating the distribution function F at a single value, the integral clearly vanishes. To find the probability of a single value, F must be discrete thereby allowing us to compute $$F(a) = sum_{a in X} p_j(a).$$ By taking the limit on $int_a^bf(x)text{dx}$ to F(a + $epsilon$) - F(a - $epsilon$) = $int_{a-epsilon}^{a+epsilon}f(x)text{dx}$, we then thus find the relative probability of F(a) occuring.






      share|cite|improve this answer









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        3 Answers
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        3 Answers
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        $begingroup$

        I would first say that you should try to be OK with it not having a super down-to-earth meaning. It can sometimes help to let yourself be abstract and use the mathematical tools you have to get somewhere you might not have gotten by purely reasoning through with statements you might use to describe other things in your daily life. On the other hand, it is often helpful to have somewhat concrete ways of thinking about things that are abstract, so it's good that you're investigating this.



        But as the PDF is the derivative of the CDF, the value of PDF $p(x)$ at $x$ is the (slope of the) best linear approximation of the CDF near $x$, i.e., that if $epsilon$ is super small, then if you want to approximate the probability that an event between $x$ and $x+epsilon$ happens linearly in $epsilon$, your best bet is to say $epsilon cdot p(x)$.






        share|cite|improve this answer










        New contributor




        csprun is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.






        $endgroup$


















          3












          $begingroup$

          I would first say that you should try to be OK with it not having a super down-to-earth meaning. It can sometimes help to let yourself be abstract and use the mathematical tools you have to get somewhere you might not have gotten by purely reasoning through with statements you might use to describe other things in your daily life. On the other hand, it is often helpful to have somewhat concrete ways of thinking about things that are abstract, so it's good that you're investigating this.



          But as the PDF is the derivative of the CDF, the value of PDF $p(x)$ at $x$ is the (slope of the) best linear approximation of the CDF near $x$, i.e., that if $epsilon$ is super small, then if you want to approximate the probability that an event between $x$ and $x+epsilon$ happens linearly in $epsilon$, your best bet is to say $epsilon cdot p(x)$.






          share|cite|improve this answer










          New contributor




          csprun is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.






          $endgroup$
















            3












            3








            3





            $begingroup$

            I would first say that you should try to be OK with it not having a super down-to-earth meaning. It can sometimes help to let yourself be abstract and use the mathematical tools you have to get somewhere you might not have gotten by purely reasoning through with statements you might use to describe other things in your daily life. On the other hand, it is often helpful to have somewhat concrete ways of thinking about things that are abstract, so it's good that you're investigating this.



            But as the PDF is the derivative of the CDF, the value of PDF $p(x)$ at $x$ is the (slope of the) best linear approximation of the CDF near $x$, i.e., that if $epsilon$ is super small, then if you want to approximate the probability that an event between $x$ and $x+epsilon$ happens linearly in $epsilon$, your best bet is to say $epsilon cdot p(x)$.






            share|cite|improve this answer










            New contributor




            csprun is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.






            $endgroup$



            I would first say that you should try to be OK with it not having a super down-to-earth meaning. It can sometimes help to let yourself be abstract and use the mathematical tools you have to get somewhere you might not have gotten by purely reasoning through with statements you might use to describe other things in your daily life. On the other hand, it is often helpful to have somewhat concrete ways of thinking about things that are abstract, so it's good that you're investigating this.



            But as the PDF is the derivative of the CDF, the value of PDF $p(x)$ at $x$ is the (slope of the) best linear approximation of the CDF near $x$, i.e., that if $epsilon$ is super small, then if you want to approximate the probability that an event between $x$ and $x+epsilon$ happens linearly in $epsilon$, your best bet is to say $epsilon cdot p(x)$.







            share|cite|improve this answer










            New contributor




            csprun is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.









            share|cite|improve this answer



            share|cite|improve this answer








            edited 1 hour ago





















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            answered 2 hours ago









            cspruncsprun

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            csprun is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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                1












                $begingroup$

                The value of the PDF is sort of like the "weighting" that each event gets for the CDF.






                share|cite|improve this answer








                New contributor




                Jacob Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.






                $endgroup$


















                  1












                  $begingroup$

                  The value of the PDF is sort of like the "weighting" that each event gets for the CDF.






                  share|cite|improve this answer








                  New contributor




                  Jacob Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.






                  $endgroup$
















                    1












                    1








                    1





                    $begingroup$

                    The value of the PDF is sort of like the "weighting" that each event gets for the CDF.






                    share|cite|improve this answer








                    New contributor




                    Jacob Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.






                    $endgroup$



                    The value of the PDF is sort of like the "weighting" that each event gets for the CDF.







                    share|cite|improve this answer








                    New contributor




                    Jacob Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.









                    share|cite|improve this answer



                    share|cite|improve this answer






                    New contributor




                    Jacob Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.









                    answered 2 hours ago









                    Jacob SmithJacob Smith

                    111




                    111




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                    New contributor





                    Jacob Smith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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                        1












                        $begingroup$


                        Definition (Gut, 2005): A distribution function F is




                        • discrete iff for some countable set of numbers {$x_j$} and point masses {$p_j$}, $$F(x) = sum_{x_jleq x} p_j, quad text{for all x} in Bbb R.$$ The function p is called the probability function.

                        • continuous iff it is continuous for all x.

                        • absolutely continuous iff there exists a non-negative, Lebesgue integrable function f, such that $$F(b) - F(a) = int_a^b f(x)text{ dx} quad {for space all space x < b.}$$ The function f is called the density of F.

                        • singular iff F $neq$ 0, F' exists and equals 0 a.e.




                        By the definition, evaluating a probability distribution function F is simply an integration of the density of said distribution. As mentioned, in the case for a $<$ b, this gives us the probability for a given range of values.



                        However, if evaluating the distribution function F at a single value, the integral clearly vanishes. To find the probability of a single value, F must be discrete thereby allowing us to compute $$F(a) = sum_{a in X} p_j(a).$$ By taking the limit on $int_a^bf(x)text{dx}$ to F(a + $epsilon$) - F(a - $epsilon$) = $int_{a-epsilon}^{a+epsilon}f(x)text{dx}$, we then thus find the relative probability of F(a) occuring.






                        share|cite|improve this answer









                        $endgroup$


















                          1












                          $begingroup$


                          Definition (Gut, 2005): A distribution function F is




                          • discrete iff for some countable set of numbers {$x_j$} and point masses {$p_j$}, $$F(x) = sum_{x_jleq x} p_j, quad text{for all x} in Bbb R.$$ The function p is called the probability function.

                          • continuous iff it is continuous for all x.

                          • absolutely continuous iff there exists a non-negative, Lebesgue integrable function f, such that $$F(b) - F(a) = int_a^b f(x)text{ dx} quad {for space all space x < b.}$$ The function f is called the density of F.

                          • singular iff F $neq$ 0, F' exists and equals 0 a.e.




                          By the definition, evaluating a probability distribution function F is simply an integration of the density of said distribution. As mentioned, in the case for a $<$ b, this gives us the probability for a given range of values.



                          However, if evaluating the distribution function F at a single value, the integral clearly vanishes. To find the probability of a single value, F must be discrete thereby allowing us to compute $$F(a) = sum_{a in X} p_j(a).$$ By taking the limit on $int_a^bf(x)text{dx}$ to F(a + $epsilon$) - F(a - $epsilon$) = $int_{a-epsilon}^{a+epsilon}f(x)text{dx}$, we then thus find the relative probability of F(a) occuring.






                          share|cite|improve this answer









                          $endgroup$
















                            1












                            1








                            1





                            $begingroup$


                            Definition (Gut, 2005): A distribution function F is




                            • discrete iff for some countable set of numbers {$x_j$} and point masses {$p_j$}, $$F(x) = sum_{x_jleq x} p_j, quad text{for all x} in Bbb R.$$ The function p is called the probability function.

                            • continuous iff it is continuous for all x.

                            • absolutely continuous iff there exists a non-negative, Lebesgue integrable function f, such that $$F(b) - F(a) = int_a^b f(x)text{ dx} quad {for space all space x < b.}$$ The function f is called the density of F.

                            • singular iff F $neq$ 0, F' exists and equals 0 a.e.




                            By the definition, evaluating a probability distribution function F is simply an integration of the density of said distribution. As mentioned, in the case for a $<$ b, this gives us the probability for a given range of values.



                            However, if evaluating the distribution function F at a single value, the integral clearly vanishes. To find the probability of a single value, F must be discrete thereby allowing us to compute $$F(a) = sum_{a in X} p_j(a).$$ By taking the limit on $int_a^bf(x)text{dx}$ to F(a + $epsilon$) - F(a - $epsilon$) = $int_{a-epsilon}^{a+epsilon}f(x)text{dx}$, we then thus find the relative probability of F(a) occuring.






                            share|cite|improve this answer









                            $endgroup$




                            Definition (Gut, 2005): A distribution function F is




                            • discrete iff for some countable set of numbers {$x_j$} and point masses {$p_j$}, $$F(x) = sum_{x_jleq x} p_j, quad text{for all x} in Bbb R.$$ The function p is called the probability function.

                            • continuous iff it is continuous for all x.

                            • absolutely continuous iff there exists a non-negative, Lebesgue integrable function f, such that $$F(b) - F(a) = int_a^b f(x)text{ dx} quad {for space all space x < b.}$$ The function f is called the density of F.

                            • singular iff F $neq$ 0, F' exists and equals 0 a.e.




                            By the definition, evaluating a probability distribution function F is simply an integration of the density of said distribution. As mentioned, in the case for a $<$ b, this gives us the probability for a given range of values.



                            However, if evaluating the distribution function F at a single value, the integral clearly vanishes. To find the probability of a single value, F must be discrete thereby allowing us to compute $$F(a) = sum_{a in X} p_j(a).$$ By taking the limit on $int_a^bf(x)text{dx}$ to F(a + $epsilon$) - F(a - $epsilon$) = $int_{a-epsilon}^{a+epsilon}f(x)text{dx}$, we then thus find the relative probability of F(a) occuring.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 1 hour ago









                            Victoria MVictoria M

                            9113




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                                19世紀

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                                この記事は検証可能な参考文献や出典が全く示されていないか、不十分です。 出典を追加して記事の信頼性向上にご協力ください。 （ 2012年3月 ） 千年紀: 2千年紀 世紀: 18世紀 - 19世紀 - 20世紀 十年紀: 1800年代 1810年代 1820年代 1830年代 1840年代 1850年代 1860年代 1870年代 1880年代 1890年代 19世紀に君臨した大英帝国。 19世紀 （じゅうきゅうせいき）は、西暦1801年から西暦1900年までの100年間を指す世紀。 目次 1 19世紀の歴史 1.1 国民国家の成立 1.2 帝国主義の興隆 1.3 列強の植民地争奪戦 2 できごと 2.1 1800年代 2.2 1810年代 2.3 1820年代 2.4 1830年代 2.5 1840年代 2.6 1850年代 2.7 1860年代 2.8 1870年代 2.9 1880年代 2.10 1890年代 2.11 1900年代 3 文化 3.1 文学 3.2 音楽 3.3 思想 3.4 科学 3.5 技術 4 人物 4.1 ヨーロッパ 4.1.1 政治家・王族 4.1.1.1 フランス 4.1.1.2 オーストリア＝ハンガリー 4.1.1.3 ロシア 4.1.1.4 イギリス 4.1.1.5 ドイツ（プロイセンほかドイツ領邦を含む） 4.1.1.6 北欧 4.1.1.7 イタリア 4.1.1.8 スペイン 4.1.1.9 ベルギー 4.1.1.10 ギリシア 4.1.2 軍人 4.1.3 実業家 4.1.4 科学と技術 4.1.5 思想と哲学・人文諸学 4.1.6 宗教 4.1.7 文学 4.1.8 美術 4.1.9 音楽 4.1.10 社会事業家 4.1.11 探検家・旅行家 4.1.12 料...
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