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Relationship between Var(XYZ) and Var(X) Var(Y) and Var(Z)

November 9, 2012

Somebody just typed the following into google to get to this blog:

relationship between var(xyz) and var(x) var(y) and var(z)

I write a lot here about moments of three variables, so I thought I’d give the answer to this question. The two variable version is easier.

First thing, the three variable problem involves coskewness…my definition of coskewness.

Second thing, if the three variables are independent and have mean zero, then the simplest intuitive thing happens,

\mathrm{Var(XYZ) = Var(X)Var(Y)Var(Z)}

But in general its much more complicated,

\mathrm{Var(XYZ) = }
\mathrm{Coskew(X^2,Y^2,Z^2) + }
\mathrm{(Var(X) + E^2(X))(Var(Y) + E^2(Y))(Var(Z) + E^2(Z)) + }
\mathrm{(Var(X) + E^2(X)) Cov(Y^2,Z^2) + }
\mathrm{(Var(Y) + E^2(Y)) Cov(X^2,Z^2) + }
\mathrm{(Var(Z) + E^2(Z)) Cov(X^2,Y^2) - }
\mathrm{(Coskew(X,Y,Z) + E(X)E(Y)E(Z) + E(X)Cov(Y,Z) + E(Y)Cov(X,Z) + E(Z)Cov(X,Y))^2}

This expression can be derived by substituting 3 identities into each other,

\mathrm{Var(X) = E(X^2) - E^2(X)}
\mathrm{Cov(X,Y) = E(XY) - E(X)E(Y)}
\mathrm{Coskew(X, Y, Z) = E(XYZ) - E(X)E(Y)E(Z) - E(X)cov(Y,Z) - E(Y)cov(X,Z) - E(Z)cov(X,Y)}

The first step to the derivation is to use the first identity to get this:

\mathrm{Var(XYZ) = E(X^2Y^2Z^2) - E^2(XYZ)}

SPECIAL CASES

If the three variables are independent, this means we can get rid of every covariance and coskewness, so we have,

\mathrm{Var(XYZ) = }
\mathrm{(Var(X) + E^2(X))(Var(Y) + E^2(Y))(Var(Z) + E^2(Z)) - E^2(X)E^2(Y)E^2(Z) = }
\mathrm{E(X^2)E(Y^2)E(Z^2) - E^2(X)E^2(Y)E^2(Z)}

Now if in addition to independence we also have zero means, we get the obvious thing,

\mathrm{Var(XYZ) = Var(X)Var(Y)Var(Z)}

Hope I didn’t screw anything up!

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