Variance problem

gwynney66

New member
Joined
Aug 14, 2018
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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.


 
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)
Yes, your expansion is correct. All three of those red expressions are equivalent.

Let's look at the last one:

X - μx - Y + μy

The Associative Property allows us to group the first two terms.

( X - μx ) - Y + μy

In order to Associate the last two terms and subtract that group, we will need to subtract -μy

Remember from algebra: subtracting a negative is the same as adding its opposite.

( X - μx ) - ( Y - μy )

Subtracting Y gives - Y and subtracting -μy gives + μy so this grouped expression is the same as what we started with.

X - μx - Y + μy
 
Thank You

Yes, your expansion is correct. All three of those red expressions are equivalent.

Let's look at the last one:

X - μx - Y + μy

The Associative Property allows us to group the first two terms.

( X - μx ) - Y + μy

In order to Associate the last two terms and subtract that group, we will need to subtract -μy

Remember from algebra: subtracting a negative is the same as adding its opposite.

( X - μx ) - ( Y - μy )

Subtracting Y gives - Y and subtracting -μy gives + μy so this grouped expression is the same as what we started with.

X - μx - Y + μy


Thanks, that makes perfect sense now, in fact I feel a bit stupid for not knowing. Like I said I just gotta try and make it through this one maths course so this probably won't be my last post.
 
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