 # What is Var(X - Y)? We know that the formula for Alan Smith 2022-01-14 Answered
What is Var(X - Y)?
We know that the formula for Var(X + Y) is Var(X) + Var(Y) + 2Cov(X,Y)
Does this mean that for Var(X - Y) it is just:
Var(X) - Var(Y) - 2Cov(X,Y)?
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$Var\left(X\right)=E\left[{X}^{2}\right]-E{\left[X\right]}^{2}$
The definition of variance.
$Var\left(X-Y\right)=E\left[{\left(X-Y\right)}^{2}\right]-E{\left[X-Y\right]}^{2}E\left[{X}^{2}-2XY+{Y}^{2}\right]-E{\left[X-Y\right]}^{2}$
Linearity of expectation:
$E\left[{X}^{2}-2XY+{Y}^{2}\right]=E\left[{X}^{2}\right]+E\left[{Y}^{2}\right]-2E\left[XY\right]\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}E\left[X-Y\right]=E\left[X\right]-E\left[Y\right]$
$Var\left(X-Y\right)=E\left[{X}^{2}\right]-2E\left[XY\right]+E\left[{Y}^{2}\right]-\left(E{\left[X\right]}^{2}-2E\left[X\right]E\left[Y\right]+E{\left[Y\right]}^{2}\right)E\left[{X}^{2}\right]-E{\left[X\right]}^{2}+E\left[{Y}^{2}\right]-E{\left[Y\right]}^{2}-2\left(E\left[XY\right]-E\left[X\right]E\left[Y\right]\right)$
Now note that:
$Cov\left(x,y\right)=E\left[\left(x-E\left[x\right]\right)\right]\left(y-E\left[y\right]\right)\right]$
$=E\left[xy\right]-E\left[xE\left[y\right]\right]-E\left[y\left[E\left[x\right]\right]+E\left[x\right]E\left[y\right]$
$=E\left[xy\right]-E\left[x\right]E\left[y\right]-E\left[y\right]E\left[x\right]+E\left[x\right]E\left[y\right]$
$=E\left[xy\right]-E\left[x\right]E\left[y\right]$
Which immediately gives the result desired in terms of the covariance.
###### Not exactly what you’re looking for? Marcus Herman
It will be Var(X)+Var(Y)−2Cov(X,Y), because Var(−Y)=Var(Y).