An Alternative Variance Formula

[mathman]

TL;DR

Interesting formula for variance

I derived (trivially) an expression for the variance of a random variable (which I had never noticed before). Let $X$ be a random variable with cdf $F(x)$ then (assuming finite second moment). $Var(X)=\frac{1}{2}\int\int (x-y)^2dF(x)dF(y)$.

Is this expression of any use?

 

[FactChecker]

It looks so much more complicated than other equations that I can't think of a use for it.
This seems much more convenient: $Var(X) = \int {X^2}dF - (\int X dF)^2$

PS. I have not taken time to verify your formula. I do not see it immediately.

 

[mathman]

It looks so much more complicated than other equations that I can't think of a use for it.
This seems much more convenient: $Var(X) = \int {X^2}dF - (\int X dF)^2$

PS. I have not taken time to verify your formula. I do not see it immediately.

$(x-y)^2=x^2+y^2-2xy$ gives the usual formula.

 

[Office_Shredder]

I think it's pretty neat.

 

[Stephen Tashi]

Is this expression of any use?

The focus in practical statistics is on estimators of populatiom parameters and sample statistics rather than the population parameters themselves. Since the formula relies on the model that we are taking independent random samples, perhaps it can motivate a statistic to test whether we actually have independent random samples. The term $(x-y)$ comparing different samples suggests the result would be influenced by any correlations. Perhaps there are already well-known statistics that we can understand to be motivated by the formula.

 

[Office_Shredder]

Is there a natural cubic expression to follow up with? You have $xF(x)$ for expected values, $\frac{1}{2}(x-y)^2F(x)F(y)$ for variance, seems like there should be a nice $\frac{1}{6}P(x,y,z)F(x)F(y)F(z)$ term for some cubic polynomial P to calculate a meaningful third moment term.