Re: symbolic Variance - Integration
- To: mathgroup at smc.vnet.net
- Subject: [mg84593] Re: [mg84585] symbolic Variance - Integration
- From: Bob Hanlon <hanlonr at cox.net>
- Date: Sun, 6 Jan 2008 05:53:58 -0500 (EST)
- Reply-to: hanlonr at cox.net
Remove[a, eV, X, Y]; Integrate[f[x] + g[x], x] // Distribute Integrate[f[x], x] + Integrate[g[x], x] eV /: eV[x_?NumericQ] := x; eV /: eV[x_ + y_] := eV[x] + eV[y]; eV /: eV[a_?NumericQ * x_] := a * eV[x]; eV /: eV[x_ * y_] := eV[x]*eV[y] + coVar[x, y]; coVar /: coVar[x_, x_] := var[x]; var /: var[x_] := eV[x^2] - eV[x]^2; indep = coVar[__] :> 0; a /: NumericQ[a] = True; var[a*X*Y] // Simplify a^2*(coVar[X^2, Y^2] + eV[X^2]*eV[Y^2]) - a^2*(coVar[X, Y] + eV[X]*eV[Y])^2 var[a*X*Y] /. indep // Simplify (-a^2)*(eV[X]^2*eV[Y]^2 - eV[X^2]*eV[Y^2]) Bob Hanlon ---- jess <cuaxie at gmail.com> wrote: > Hi there! > > I came across a formula for variance of a product of random variables > X and Y. The formula was given under the assumption that X and Y are > independent, and there was a remark that without this assumption the > formula is very complicated. > > I thought to myslef, okay i have mathematica on my pc so let's see how > complicated it is... > > Then i realized that the only way to calculate variance in mathematica > is to: > - give a list of points (i.e. uniform discrete distribution) > - give a specific distribution, and say, its parameters ( eg. normal > or gamma dist.) > > so there is no way to make a symbolic calculation using an arbitrary > random variable... > > Huuuh, so i thought okay variance and expected value are nothing but > integrals so i can try to define my own function using integrals (i do > not want to consider just discrete or continues random variables)... > > and then i realized i can't even obtain that integration is additive > i.e. from > > Integrate[f[x] + g[x], x] > > i could not get > > Integrate[f[x], x] + Integrate[g[x], x] > > (so my function for expected value would not even tell me that > EValue(X+Y) = EValue(X) + EValue(Y)... not to mention about > calculating Variance( XY) = ????.... ) > > Coming back to my original problem: how to make mathematica do this > > In[1] =E[XY] > Out[1]=E[X]E[Y]+Cov[XY] > > and further > > In[2]=Var[XY] > In[2]= ?????????? > > Please I would appreciate absolutely any comment at all... > > Jess >