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I am trying to find the variance of $u$.

I know that if:

$u = a \cdot X + b \cdot Y$

Then:

$var(u) = a^2 \cdot var(X) + b^2 \cdot var(Y) + 2\cdot a\cdot b \cdot cov(X,Y)$

The problem I'm trying to solve is to find $var(u)$ when:

$u = X(a-Y)$

Where $a$ is a constant and $X$ and $Y$ are random (normally distributed) variables.

SOLUTION

Thanks to all who helped. I have managed to solve this. For the benefit of others facing this problem in the future, here is my solution:

$u = X(a-Y)$

Expand:

$u = X \cdot a - X \cdot Y$

The Wikipedia page for the Delta Method gives the following formula:

$Var(h_r) = \sum_{i}\left(\frac{\partial h_r}{\partial B_i}\right)^2 Var(B_i) + \sum_{i}\sum_{j\neq i} \left(\frac{\partial h_r}{\partial B_i}\right)\left(\frac{\partial h_r}{\partial B_j}\right)Cov \left(B_i,B_j\right)$

So I need to calculate the partial derivatives with respect to each variable:

$\frac{\partial u}{\partial X}=a-Y$

$\frac{\partial u}{\partial Y}=-X$

Putting these into the formula gives:

$Var(u)=(a-Y)^2.Var(X) + (-X)^2.Var(Y)+2.(a-Y).(-X).Cov(X,Y)$

Simplifying slightly:

$Var(u)=(a-Y)^2.Var(X) + X^2.Var(Y)-2.X.(a-Y).Cov(X,Y)$

Thanks again for your pointers. Hopefully someone finds this useful one day.

Matthew Reid
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    Look at $U = aX - XY.$ Use definition of $Cov(X,Y).$ – BruceET May 04 '20 at 05:55
  • Further hint: since $(X,a-Y)$ also has a bivariate Normal distribution, it suffices to solve the problem when $U=XY.$ That solution is given at https://stats.stackexchange.com/questions/15978/variance-of-product-of-dependent-variables. – whuber May 04 '20 at 11:04
  • Thanks for the hints, but I've been looking at this for 2 days and I'm thoroughly stuck. I know how to solve U=XY but how do I solve U=aX - XY ? A few lines of working out would be really appreciated. – Matthew Reid May 04 '20 at 11:18
  • If you can solve the case $U=XY$ you don't have to do anything at all, because--as I pointed out--there is absolutely no difference in the problems and there's nothing to work out. You only need to know how to relate the parameters of $a-Y$ to those of $Y.$ – whuber May 04 '20 at 11:23

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