Integral over an interval of the normal distribution












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$begingroup$


Assume that $f$ is the normal distribution function with mean equal to $0$ and variance equal to $sigma^2$.



Set some interval $I=[a,b]$ in $mathbb{R}^+$.



Denote the integral of $f$ over $I$ by $S_I(sigma^2)$.



My question: is $S_I(sigma^2)$ monotonous (decreasing?) as $sigma^2$ increases continuously? Or is it dependent on $I$ or the initial value of $sigma^2$?



Thank you!










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    0












    $begingroup$


    Assume that $f$ is the normal distribution function with mean equal to $0$ and variance equal to $sigma^2$.



    Set some interval $I=[a,b]$ in $mathbb{R}^+$.



    Denote the integral of $f$ over $I$ by $S_I(sigma^2)$.



    My question: is $S_I(sigma^2)$ monotonous (decreasing?) as $sigma^2$ increases continuously? Or is it dependent on $I$ or the initial value of $sigma^2$?



    Thank you!










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Assume that $f$ is the normal distribution function with mean equal to $0$ and variance equal to $sigma^2$.



      Set some interval $I=[a,b]$ in $mathbb{R}^+$.



      Denote the integral of $f$ over $I$ by $S_I(sigma^2)$.



      My question: is $S_I(sigma^2)$ monotonous (decreasing?) as $sigma^2$ increases continuously? Or is it dependent on $I$ or the initial value of $sigma^2$?



      Thank you!










      share|cite|improve this question









      $endgroup$




      Assume that $f$ is the normal distribution function with mean equal to $0$ and variance equal to $sigma^2$.



      Set some interval $I=[a,b]$ in $mathbb{R}^+$.



      Denote the integral of $f$ over $I$ by $S_I(sigma^2)$.



      My question: is $S_I(sigma^2)$ monotonous (decreasing?) as $sigma^2$ increases continuously? Or is it dependent on $I$ or the initial value of $sigma^2$?



      Thank you!







      probability






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      asked Jan 16 at 12:07









      DaugmentedDaugmented

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          $begingroup$

          Consider this plot and the interval around $2$. It's clear that the integral is not monotonous with $sigma$. So in general case, it depends both on interval (for symmetric or wide enough intervals, the integral is monotonous) and on initial value of $sigma^2$ (for each interval there is such $sigma_0$, so for $sigma>sigma_0$ the integral is monotonous).



          enter image description here






          share|cite|improve this answer









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            1












            $begingroup$

            Consider this plot and the interval around $2$. It's clear that the integral is not monotonous with $sigma$. So in general case, it depends both on interval (for symmetric or wide enough intervals, the integral is monotonous) and on initial value of $sigma^2$ (for each interval there is such $sigma_0$, so for $sigma>sigma_0$ the integral is monotonous).



            enter image description here






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              Consider this plot and the interval around $2$. It's clear that the integral is not monotonous with $sigma$. So in general case, it depends both on interval (for symmetric or wide enough intervals, the integral is monotonous) and on initial value of $sigma^2$ (for each interval there is such $sigma_0$, so for $sigma>sigma_0$ the integral is monotonous).



              enter image description here






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                Consider this plot and the interval around $2$. It's clear that the integral is not monotonous with $sigma$. So in general case, it depends both on interval (for symmetric or wide enough intervals, the integral is monotonous) and on initial value of $sigma^2$ (for each interval there is such $sigma_0$, so for $sigma>sigma_0$ the integral is monotonous).



                enter image description here






                share|cite|improve this answer









                $endgroup$



                Consider this plot and the interval around $2$. It's clear that the integral is not monotonous with $sigma$. So in general case, it depends both on interval (for symmetric or wide enough intervals, the integral is monotonous) and on initial value of $sigma^2$ (for each interval there is such $sigma_0$, so for $sigma>sigma_0$ the integral is monotonous).



                enter image description here







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 16 at 12:37









                Vasily MitchVasily Mitch

                2,6791312




                2,6791312






























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