hobz Posted June 30, 2011 Share Posted June 30, 2011 (edited) 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, 2011 by hobz Link to comment Share on other sites More sharing options...

baxtrom Posted June 30, 2011 Share Posted June 30, 2011 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. Link to comment Share on other sites More sharing options...

hobz Posted June 30, 2011 Author Share Posted June 30, 2011 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. Link to comment Share on other sites More sharing options...

mathematic Posted July 1, 2011 Share Posted July 1, 2011 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. Link to comment Share on other sites More sharing options...

DrRocket Posted July 1, 2011 Share Posted July 1, 2011 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. Link to comment Share on other sites More sharing options...

## Recommended Posts

## Create an account or sign in to comment

You need to be a member in order to leave a comment

## Create an account

Sign up for a new account in our community. It's easy!

Register a new account## Sign in

Already have an account? Sign in here.

Sign In Now