How do you DERIVE the BELL CURVE?
Why Cross Validated Stack Exchange recommends it
Stats Stack Exchange (Cross Validated) users link it when asked how the normal distribution can be derived or discovered, and as the source for dartboard-derivation questions.
Recommended by 1 people across 1 different months, 2018 to 2020.
What it is
Mathoma derives the normal (Gaussian) probability density from first principles using a dartboard argument, assuming the x and y errors are independent and that the density depends only on distance from the center. The approach follows Herschel (1850) and Maxwell (1860) and ends with a functional equation and a Gaussian integral. It runs about 35 minutes with references listed in the description.
What people said
Strong rec Sebastian · 2018-02 · Cross Validated Stack Exchange · on “How could I have discovered the normal distribution?”
I also asked myself that question and this youtube video is the best answer I have found [this video] I don`t think that it is the original derivation but the description of the video says "This argument is adapted from the work of the astronomer John Herschel in 1850 and the physicist James Clerk Maxwell in 1860. "
Cites frt132 · 2018-08 · Cross Validated Stack Exchange · on “The assumptions in the derivation of normal distribution using a dart board example?”
The derivation could be found here: [LINK] or here [this video] They assume that (1) the pdf f(x), f(y) are independent from each other and (2) the pdf of a point g(x,y) only depends on the distance of the point to the bull's eye. There are definitely examples of pdfs that do not satisfy both (1) and (2). For example, imagine a discrete example where the dart landing at (2,1) or (1,2) both have a probability of 0.4 and the dart landing at (2,2) has a probability of 0.2. This satisfy the assumption (2) that the probability depends only on the distance but not (1). What kind of pdfs can satisfy both (1) and (2)?
Cites Ryan Volpi · 2020-07 · Cross Validated Stack Exchange · on “Connection between derivation of normal distribution and the central limit theorem”
I was under the impression that the reason that the normal distribution occurs naturally could be explained by the central limit theorem (CLT). I recently watched a video [this video] that described a derivation of the normal probability density function using the case of darts clustered around the center of a dartboard. As far as I can tell, the example is distinct from the CLT. The assumptions about the distribution of darts are as follows: Darts are clustered around a center and are less dense further from the center. The distribution of darts in the X direction is statistically independent of the distribution of darts in the Y direction. The distribution of darts is equivalent for any point w