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]
[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]