Hi so I just have a minor issue with an example in my econometrics text book surrounding variance.
So the example is algebraically proving that:
var(X-Y) = var(X) + var(Y) - 2cov(X, Y)
It has stated to let Z = X-Y
Then calculated the Expected Value:
E(Z) = μz = E(X) - E(Y) = μx - μy
Then it launches into the variance calculations, I'm gonna post it in full and then just point out what step I'm having issues on:
var(X-Y) = var(Z) = E[(Z - μz)2]
= E[(X - Y - (μx - μy))2]
= E{ [(X - μx) - (Y - μy)]2 }
= E[(X - μx)2 + (Y - μy)2 - 2(X - μx) - (Y - μy)]
= E[(X - μx)2] + E[(Y - μy)2] - 2E[(X - μx)(Y - μy)]
= var(X) + var(Y) - 2cov(X, Y)
So essentially I'm unsure why (X - Y - (μx - μy)) moves to (X - μx) - (Y - μy) my understanding was that you would expand the negative to inside the brackets and get essentially (X - μx - Y + μy)
Sorry if this is something that may be very basic, my degree requires one year of econometrics despite the fact I want to do nothing even close to that.
I hope someone can give me a hand with this.
So the example is algebraically proving that:
var(X-Y) = var(X) + var(Y) - 2cov(X, Y)
It has stated to let Z = X-Y
Then calculated the Expected Value:
E(Z) = μz = E(X) - E(Y) = μx - μy
Then it launches into the variance calculations, I'm gonna post it in full and then just point out what step I'm having issues on:
var(X-Y) = var(Z) = E[(Z - μz)2]
= E[(X - Y - (μx - μy))2]
= E{ [(X - μx) - (Y - μy)]2 }
= E[(X - μx)2 + (Y - μy)2 - 2(X - μx) - (Y - μy)]
= E[(X - μx)2] + E[(Y - μy)2] - 2E[(X - μx)(Y - μy)]
= var(X) + var(Y) - 2cov(X, Y)
So essentially I'm unsure why (X - Y - (μx - μy)) moves to (X - μx) - (Y - μy) my understanding was that you would expand the negative to inside the brackets and get essentially (X - μx - Y + μy)
Sorry if this is something that may be very basic, my degree requires one year of econometrics despite the fact I want to do nothing even close to that.
I hope someone can give me a hand with this.