Derivation of standard distribution probability density function

[keng]

I'm trying to learn to derive the pdf of normal distribution, that is
[FONT=MathJax_Math-italic]f[FONT=MathJax_Main]([FONT=MathJax_Math-italic]x[FONT=MathJax_Main];[FONT=MathJax_Math-italic]μ[FONT=MathJax_Main],[FONT=MathJax_Math-italic]σ[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1[FONT=MathJax_Math-italic]σ[FONT=MathJax_Main]2[FONT=MathJax_Main]√[FONT=MathJax_Math-italic]π[/FONT][/FONT][/FONT][/FONT][/FONT][FONT=MathJax_Math-italic]e[FONT=MathJax_Main]−[FONT=MathJax_Main]1[FONT=MathJax_Main]2[FONT=MathJax_Main]([FONT=MathJax_Math-italic]x[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]μ[/FONT][FONT=MathJax_Math-italic]σ[/FONT][/FONT][/FONT][FONT=MathJax_Main])[FONT=MathJax_Main]2[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT]
[/FONT][/FONT]
Firstly, there is a hypothesis to derive this pdf, which is the coordinate is farther away from the origin, the lower the value of [FONT=MathJax_Math-italic]f[FONT=MathJax_Main]([FONT=MathJax_Math-italic]x[FONT=MathJax_Main])[/FONT][/FONT][/FONT].[/FONT]
After defining the area and transformation to polar form on the cartesian plane, I get [FONT=MathJax_Math-italic]f[FONT=MathJax_Main]([FONT=MathJax_Math-italic]x[FONT=MathJax_Main])[FONT=MathJax_Main]=[FONT=MathJax_Math-italic]A[FONT=MathJax_Math-italic]e[FONT=MathJax_Math-italic]C[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]2[FONT=MathJax_Main]2[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT].[/FONT]
By the hypothesis, the C of [FONT=MathJax_Math-italic]f[FONT=MathJax_Main]([FONT=MathJax_Math-italic]x[FONT=MathJax_Main])[FONT=MathJax_Main]=[FONT=MathJax_Math-italic]A[FONT=MathJax_Math-italic]e[FONT=MathJax_Math-italic]C[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]2[FONT=MathJax_Main]2[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT] will be negative. So the question is what is the theorem that implies C is negative.[/FONT][/FONT][/FONT][/FONT][/FONT]

 

[j-astron]

Hi keng,

Unfortunately, the formatting of your equations didn't survive the process of copying them from wherever you got them from. I've replaced them in the quotes below with what I think/guess you meant:

I'm trying to learn to derive the pdf of normal distribution, that is \(\displaystyle f(x;\mu,\sigma^2)=\dfrac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \)

Firstly, there is a hypothesis to derive this pdf, which is the coordinate is farther away from the origin, the lower the value of \(\displaystyle f(x)\)

Umm, wouldn't there be many functional forms that satisfied this criterion? How is that statement alone going to allow you to arrive at a Gaussian specifically?

After defining the area and transformation to polar form on the cartesian plane, I get \(\displaystyle f(x)=Ae^{\frac{Cx^2}{2}}\).

Again, the above equation is what I guessed you were trying to type. If my guess is correct, then I'm super curious how you got so close to the required functional form based on such a vague criterion. Can you please post your work, including the polar transformation, etc?