mathematical curiosity (one of those again)

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  • iCeCuBEz v2
    XFD
    • Mar 2008
    • 4924
    #1

    mathematical curiosity (one of those again)

    OK, so in my probability and statistics class we talked about normal distribution, which is defined by;



    where mu is the average (or first moment about the origin) and lowercase sigma is standard deviation (or square root of variance)

    This is a probability distribution, so the area under this curve over the entire interval is going to be 1.

    OK that's easy peasy, but a while ago I remember stumbling upon this formula.... the Gaussian Integral (not to be confused with Gaussian Distribution)



    These formulas, for what they are representing are very very similar. They have the same curvature and are over the same interval.

    The discrepancy between the two is the computed integrals. The normal distribution will have an area under its curvature of 1 (obviously), and the area under the Gaussian Integral I just presented is the square root of pi. The area of the Gaussian Integral can be easily proved with this method.



    so anyway my question......

    is there any bridge between these formulas?
    Last edited by iCeCuBEz v2; 11-18-2014, 11:05 AM.
    I bring my math homework to church. It helps me find a higher power.

    Dennis, Nell, Edna, Leon, Nedra, Anita, Rolf, Nora, Alice, Carol, Leo, Jane, Reed, Dena, Dale, Basil, Rae, Penny, Lana, Dave, Denny, Lena, Ida, Bernadette, Ben, Ray, Lila, Nina, Jo, Ira, Mara, Sara, Mario, Jan, Ina, Lily, Arne, Bette, Dan, Reba, Diane, Lynn, Ed, Eva, Dana, Lynne, Pearl, Isabel, Ada, Ned, Dee, Rena, Joel, Lora, Cecil, Aaron, Flora, Tina, Arden, Noel, and Ellen sinned.
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    • Zapmeister
      FFR Player
      • Sep 2012
      • 466
      #2
      Re: mathematical curiosity (one of those again)

      not sure what question you're asking. are you trying to verify using the gaussian integral formula that the area under the normal distribution curve is indeed 1? if you start with the gaussian integral formula and substitute y = (x - mu)/(sigma*sqrt2) then the normal distribution pdf becomes 1/sqrt(pi)*exp(-y^2) which integrates to 1

      let me know if you were asking a different question

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      ask me anything here

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      final position: 4th

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        • iCeCuBEz v2
          XFD
          • Mar 2008
          • 4924
          #3
          Re: mathematical curiosity (one of those again)

          Are these formulas related.... like could one be written as the other or something... I guess that's what I'm asking

          OK THIS is what im asking, could you algebraically manipulate one of those formulas to arrive at the other one or vice versa?
          I bring my math homework to church. It helps me find a higher power.

          Dennis, Nell, Edna, Leon, Nedra, Anita, Rolf, Nora, Alice, Carol, Leo, Jane, Reed, Dena, Dale, Basil, Rae, Penny, Lana, Dave, Denny, Lena, Ida, Bernadette, Ben, Ray, Lila, Nina, Jo, Ira, Mara, Sara, Mario, Jan, Ina, Lily, Arne, Bette, Dan, Reba, Diane, Lynn, Ed, Eva, Dana, Lynne, Pearl, Isabel, Ada, Ned, Dee, Rena, Joel, Lora, Cecil, Aaron, Flora, Tina, Arden, Noel, and Ellen sinned.

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            • Dynam0
              The Dominator
              • Sep 2005
              • 8987
              #4
              Re: mathematical curiosity (one of those again)

              The normal distribution function is just a transformation of a Gaussian Function. The variance/mean/etc. all impose characteristics on this function to ensure that the integral is equal to 1.

              Zapmeister proved it but not in a detailed line-by-line way like your post did for a base Gaussian Function.

              edit:

              here is the standard form for a Gaussian Function:

              Last edited by Dynam0; 11-18-2014, 11:31 AM.

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                • iCeCuBEz v2
                  XFD
                  • Mar 2008
                  • 4924
                  #5
                  Re: mathematical curiosity (one of those again)

                  yeah I understand that I was just looking for the step by step way of doing it I just couldn't do it on my own and I didn't see one online.

                  ok dynamo that's all I needed to see question solved idk where you found that that was EXACTLY what I was looking for.....

                  wow painfully obvious I'm mad at myself for not figuring that out. the curve ball was that the parent function can't really be deduced (at least by me) by looking at e^(-x^2)
                  Last edited by iCeCuBEz v2; 11-18-2014, 11:54 AM.
                  I bring my math homework to church. It helps me find a higher power.

                  Dennis, Nell, Edna, Leon, Nedra, Anita, Rolf, Nora, Alice, Carol, Leo, Jane, Reed, Dena, Dale, Basil, Rae, Penny, Lana, Dave, Denny, Lena, Ida, Bernadette, Ben, Ray, Lila, Nina, Jo, Ira, Mara, Sara, Mario, Jan, Ina, Lily, Arne, Bette, Dan, Reba, Diane, Lynn, Ed, Eva, Dana, Lynne, Pearl, Isabel, Ada, Ned, Dee, Rena, Joel, Lora, Cecil, Aaron, Flora, Tina, Arden, Noel, and Ellen sinned.

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                    • Charles Claythorne
                      FFR Player
                      • Apr 2013
                      • 138
                      #6
                      Re: mathematical curiosity (one of those again)

                      So if I understand correctly, you just want to derive the Gaussian Integral from the Gaussian Distribution? If that's the case, it's just simply a case of a few substitutions.

                      I was trying to use LaTeX, but I kept screwing something up, so I wrote it down instead!!! The picture is sort of big though, so I apologise about that.

                      Last edited by Charles Claythorne; 11-21-2014, 02:25 PM.

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                        • stargroup100
                          behanjc & me are <3'ers
                          FFR Simfile Author
                          FFR Music Producer
                          • Jul 2006
                          • 2051
                          #7
                          Re: mathematical curiosity (one of those again)

                          One bit of insight as to how you can know these formulas can be derived from each other without even needing to do any work is knowing the basic relationships of functions and chain rule. For example, these two equations are related the same way you know that f(x) = x^2 is related to g(x) = 3(x-4)^2.

                          The transformations that turn the Gaussian Integral into the normal distribution are:
                          - Horizontal translation by a constant, μ (which obviously does not affect the integral given the limits follow the translation)
                          - Horizontal stretch by a constant factor, σ√2 (which changes the integral by that factor)
                          - Vertical stretch by a constant factor, 1/σ√(2π) (which changes the integral by that factor)

                          √π * σ√2 / σ√(2π) = 1
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                            • iCeCuBEz v2
                              XFD
                              • Mar 2008
                              • 4924
                              #8
                              Re: mathematical curiosity (one of those again)

                              Originally posted by Charles Claythorne
                              So if I understand correctly, you just want to derive the Gaussian Integral from the Gaussian Distribution? If that's the case, it's just simply a case of a few substitutions.

                              I was trying to use LaTeX, but I kept screwing something up, so I wrote it down instead!!! The picture is sort of big though, so I apologise about that.

                              thanks

                              and yes sg i understand how they are derived with translations i wanted to see the work
                              Last edited by iCeCuBEz v2; 11-23-2014, 10:40 PM.
                              I bring my math homework to church. It helps me find a higher power.

                              Dennis, Nell, Edna, Leon, Nedra, Anita, Rolf, Nora, Alice, Carol, Leo, Jane, Reed, Dena, Dale, Basil, Rae, Penny, Lana, Dave, Denny, Lena, Ida, Bernadette, Ben, Ray, Lila, Nina, Jo, Ira, Mara, Sara, Mario, Jan, Ina, Lily, Arne, Bette, Dan, Reba, Diane, Lynn, Ed, Eva, Dana, Lynne, Pearl, Isabel, Ada, Ned, Dee, Rena, Joel, Lora, Cecil, Aaron, Flora, Tina, Arden, Noel, and Ellen sinned.

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                                • JurseyRider734
                                  lil j the bad b-word
                                  • Aug 2003
                                  • 7506
                                  #9
                                  Re: mathematical curiosity (one of those again)

                                  why dont u just ask nick
                                  Originally posted by Arch0wl
                                  I'd better be considering I own roughly six textbooks on logic and have taken courses involving its extensive use

                                  Originally posted by Afrobean
                                  Originally Posted by JurseyRider734
                                  the fact that you're resorting to threatening physical violence says a lot anyway.
                                  Just that you're a piece of shit who can't see reason and instead deserves a fucking beating.

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                                    • MasterOfTheTaco
                                      deer god
                                      • Mar 2008
                                      • 377
                                      #10
                                      Re: mathematical curiosity (one of those again)

                                      ¯\_(ツ)_/¯

                                        Comment

                                        • iCeCuBEz v2
                                          XFD
                                          • Mar 2008
                                          • 4924
                                          #11
                                          Re: mathematical curiosity (one of those again)

                                          Originally posted by JurseyRider734
                                          why dont u just ask nick
                                          next time
                                          I bring my math homework to church. It helps me find a higher power.

                                          Dennis, Nell, Edna, Leon, Nedra, Anita, Rolf, Nora, Alice, Carol, Leo, Jane, Reed, Dena, Dale, Basil, Rae, Penny, Lana, Dave, Denny, Lena, Ida, Bernadette, Ben, Ray, Lila, Nina, Jo, Ira, Mara, Sara, Mario, Jan, Ina, Lily, Arne, Bette, Dan, Reba, Diane, Lynn, Ed, Eva, Dana, Lynne, Pearl, Isabel, Ada, Ned, Dee, Rena, Joel, Lora, Cecil, Aaron, Flora, Tina, Arden, Noel, and Ellen sinned.

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                                            • igotrhythm
                                              Fractals!
                                              • Sep 2004
                                              • 6535
                                              #12
                                              Re: mathematical curiosity (one of those again)

                                              Another fun thing about the normal curve is that the inflection points are located one standard deviation away from the mean/median/mode.
                                              Originally posted by thesunfan
                                              I literally spent 10 minutes in the library looking for the TWG forum on Smogon and couldn't find it what the fuck is this witchcraft IGR

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                                                • Soundwave-
                                                  Carry your failures proud
                                                  • Sep 2015
                                                  • 644
                                                  #13
                                                  Re: mathematical curiosity (one of those again)

                                                  Originally posted by igotrhythm
                                                  Another fun thing about the normal curve is that the inflection points are located one standard deviation away from the mean/median/mode.
                                                  Huh.

                                                  ...

                                                  Prove it.
                                                  Originally posted by [11:38 PM] Hakulyte
                                                  only person who can legit tilt me is like YoshL
                                                  Originally posted by スンファンさん
                                                  右に3回回らない限り間違います。

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                                                    • benguino
                                                      Kawaii Desu Ne?
                                                      • Dec 2007
                                                      • 4186
                                                      #14
                                                      Re: mathematical curiosity (one of those again)

                                                      Originally posted by Soundwave-
                                                      Huh.

                                                      ...

                                                      Prove it.
                                                      Prove one standard deviation lies on inflection points
                                                      http://math.stackexchange.com/questions/190066/prove-one-standard-deviation-lies-on-inflection-points
                                                      Is my conjecture correct that one standard deviation lies on the inflection points of the normal distribution curve (of the probability density function)? How can it be proved using the standard
                                                      AMA: http://ask.fm/benguino


                                                      Originally posted by Spenner
                                                      (^)> peck peck says the heels
                                                      Originally posted by Xx{Midnight}xX
                                                      And god made ben, and realized he was doomed to miss. And said it was good.
                                                      Originally posted by Zakvvv666
                                                      awww :< crushing my dreams; was looking foward to you attempting to shoot yourself point blank and missing

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                                                        • Jtehanonymous
                                                          Hunger Games Hunty
                                                          • Jan 2007
                                                          • 3770
                                                          #15
                                                          Re: mathematical curiosity (one of those again)

                                                          good read

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