To find the mean and variance with given conditions












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$X$ follows normal distribution $mathcal{N}(mu , sigma^2)$ with pdf $f$ and cdf $F$. If $max_x f(x)= 0.997356$ and $F(-1)+F(7)=1$, determine $mu, sigma^2$ and $mathbb{P}[Xle 0]$ .




I have no clue about this question and unable to interpret the given conditions. How can I relate the max pdf to find the mean. Even if I get to know the first part I can calculate the rest. Any help would be grateful.






probability-distributions normal-distribution







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    0












    $begingroup$



    $X$ follows normal distribution $mathcal{N}(mu , sigma^2)$ with pdf $f$ and cdf $F$. If $max_x f(x)= 0.997356$ and $F(-1)+F(7)=1$, determine $mu, sigma^2$ and $mathbb{P}[Xle 0]$ .




    I have no clue about this question and unable to interpret the given conditions. How can I relate the max pdf to find the mean. Even if I get to know the first part I can calculate the rest. Any help would be grateful.






    probability-distributions normal-distribution







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$



      $X$ follows normal distribution $mathcal{N}(mu , sigma^2)$ with pdf $f$ and cdf $F$. If $max_x f(x)= 0.997356$ and $F(-1)+F(7)=1$, determine $mu, sigma^2$ and $mathbb{P}[Xle 0]$ .




      I have no clue about this question and unable to interpret the given conditions. How can I relate the max pdf to find the mean. Even if I get to know the first part I can calculate the rest. Any help would be grateful.






      probability-distributions normal-distribution







      share|cite|improve this question











      $endgroup$





      $X$ follows normal distribution $mathcal{N}(mu , sigma^2)$ with pdf $f$ and cdf $F$. If $max_x f(x)= 0.997356$ and $F(-1)+F(7)=1$, determine $mu, sigma^2$ and $mathbb{P}[Xle 0]$ .




      I have no clue about this question and unable to interpret the given conditions. How can I relate the max pdf to find the mean. Even if I get to know the first part I can calculate the rest. Any help would be grateful.






      probability-distributions normal-distribution




      probability-distributions normal-distribution






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      share|cite|improve this question













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      edited Dec 13 '18 at 12:27









      gt6989b

      34k22455




      34k22455










      asked Dec 13 '18 at 12:20









      Kriti AroraKriti Arora

      396




      396






















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          HINT



          Since $X sim mathcal{N}left(mu,sigma^2right)$, you know that
          $$
          f(x) = frac{1}{sigma sqrt{2pi}}
          expleft(frac{-1}{2} left(frac{x-mu}{sigma} right)^2 right)
          $$
          which is the bell curve, clearly reaching maximum at $x = mu$. What is this maximum, and can you find $sigma$ from its value?



          Then use the fact that $(X-mu)/sigma sim mathcal{N}(0,1)$ and usual relationship of the std normal cdf to figure out $mu$ from the second relation.






          share|cite|improve this answer









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            $begingroup$

            HINT



            Since $X sim mathcal{N}left(mu,sigma^2right)$, you know that
            $$
            f(x) = frac{1}{sigma sqrt{2pi}}
            expleft(frac{-1}{2} left(frac{x-mu}{sigma} right)^2 right)
            $$
            which is the bell curve, clearly reaching maximum at $x = mu$. What is this maximum, and can you find $sigma$ from its value?



            Then use the fact that $(X-mu)/sigma sim mathcal{N}(0,1)$ and usual relationship of the std normal cdf to figure out $mu$ from the second relation.






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              HINT



              Since $X sim mathcal{N}left(mu,sigma^2right)$, you know that
              $$
              f(x) = frac{1}{sigma sqrt{2pi}}
              expleft(frac{-1}{2} left(frac{x-mu}{sigma} right)^2 right)
              $$
              which is the bell curve, clearly reaching maximum at $x = mu$. What is this maximum, and can you find $sigma$ from its value?



              Then use the fact that $(X-mu)/sigma sim mathcal{N}(0,1)$ and usual relationship of the std normal cdf to figure out $mu$ from the second relation.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                HINT



                Since $X sim mathcal{N}left(mu,sigma^2right)$, you know that
                $$
                f(x) = frac{1}{sigma sqrt{2pi}}
                expleft(frac{-1}{2} left(frac{x-mu}{sigma} right)^2 right)
                $$
                which is the bell curve, clearly reaching maximum at $x = mu$. What is this maximum, and can you find $sigma$ from its value?



                Then use the fact that $(X-mu)/sigma sim mathcal{N}(0,1)$ and usual relationship of the std normal cdf to figure out $mu$ from the second relation.






                share|cite|improve this answer









                $endgroup$



                HINT



                Since $X sim mathcal{N}left(mu,sigma^2right)$, you know that
                $$
                f(x) = frac{1}{sigma sqrt{2pi}}
                expleft(frac{-1}{2} left(frac{x-mu}{sigma} right)^2 right)
                $$
                which is the bell curve, clearly reaching maximum at $x = mu$. What is this maximum, and can you find $sigma$ from its value?



                Then use the fact that $(X-mu)/sigma sim mathcal{N}(0,1)$ and usual relationship of the std normal cdf to figure out $mu$ from the second relation.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 13 '18 at 12:25









                gt6989bgt6989b

                34k22455




                34k22455






























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