I just learned that the normal distribution is the solution to

[math]

\frac{\mathrm{d}y}{\mathrm{d}x}+yx = 0

[/math]

What problem led to this differential equation?

How does this equation lead people to think of distributions?

In a "real" world problem (physics) what could [math]y[/math] and [math]x[/math] be?

Edited June 30, 201114 yr by hobz

I just learned that the normal distribution is the solution to

[math]\frac{\mathrm{d}y}{\mathrm{d}x}+x = 0[/math]

What problem led to this differential equation?

How does this equation lead people to think of distributions?

In a "real" world problem (physics) what could [math]y[/math] and [math]x[/math] be?

 

 

Hi there,

as far as I can see the solution to that DE is a parabolic, [math]y(x) = -\frac{x^2}{2} + C[/math]. According to mathworld this functional DE defines the normal distribution:

 

[math] \frac{\mathrm{d}y(x)}{\mathrm{d}x} = \frac{y(\mu- x)}{\sigma^2} [/math],

 

where [math]\mu[/math] and [math]\sigma[/math] are the mean and standard deviation, respectively.

Hi there,

as far as I can see the solution to that DE is a parabolic, [math]y(x) = -\frac{x^2}{2} + C[/math]. According to mathworld this functional DE defines the normal distribution:

 

[math] \frac{\mathrm{d}y(x)}{\mathrm{d}x} = \frac{y(\mu- x)}{\sigma^2} [/math],

 

where [math]\mu[/math] and [math]\sigma[/math] are the mean and standard deviation, respectively.

 

I forgot a [math]y[/math] in the eq.

The origin of the normal distribution is the Central Limit Theorem. The differential equation happens to be an obvious derivation of the normal density function.

The origin of the normal distribution is the Central Limit Theorem. The differential equation happens to be an obvious derivation of the normal density function.

 

The origin of the normal distribution was in the work of Gauss. The distribution is also known as the Gaussian distribution.

 

Besides being the limiting density of a sum of independent identically distributed random variables, the central limit theorem, it is also distinguished by the fact that it is it's own Fourier transform and that a convolution produuct of Gaussian densities is again a Gaussian density.