Var(X+Y+Z)

shrek

New Member
#1
What is the formula for the variance of 3 dependent variables?

For 2 dependent variables, the formula is

Var(X)+Var(Y)+2*Cov(X,Y)

What is Var(X+Y+Z) if the variables are dependent? Please do not give me the general formula for n number of variables. I've seen it but I don't understand it. If I see an example with 3 variables then I think I can figure it out. Thanks.
 
Last edited:

Dragan

Super Moderator
#2
What is the formula for the variance of 3 dependent variables?

For 2 dependent variables, the formula is

Var(X)+Var(Y)+2*Cov(X,Y)

What is Var(X+Y+Z) if the variables are dependent? Please go not give me the general formula for n number of variables. I've seen it but I don't understand it. If I see an example with 3 variables then I think I can figure it out. Thanks.

How about:

Var (X + Y + Z) = Var(X) + Var(Y) + Var(Z) + 2Cov(X,Y) + 2Cov(X,Z) + 2Cov(Y,Z).


I can write out a short proof if you would like.
 

shrek

New Member
#3
How about:

Var (X + Y + Z) = Var(X) + Var(Y) + Var(Z) + 2Cov(X,Y) + 2Cov(X,Z) + 2Cov(Y,Z).


I can write out a short proof if you would like.
thanks! no proof necessary. how about Var(X+Y-Z)? is this right?

Var (X + Y - Z) = Var(X) + Var(Y) - Var(Z) + 2Cov(X,Y) - 2Cov(X,Z) - 2Cov(Y,Z)
 

Dragan

Super Moderator
#4
thanks! no proof necessary. how about Var(X+Y-Z)? is this right?

Var (X + Y - Z) = Var(X) + Var(Y) - Var(Z) + 2Cov(X,Y) - 2Cov(X,Z) - 2Cov(Y,Z)

Well, you're always going to sum the variances so we have,


Var (X + Y - Z) = Var(X) + Var(Y) + Var(Z) + 2Cov(X,Y) - 2Cov(X,Z) - 2Cov(Y,Z)
 

BGM

TS Contributor
#6
\( Var\left[\sum_{i=1}^n X_i\right] \)

\( = Cov\left[\sum_{i=1}^n X_i, \sum_{j=1}^n X_j\right] \)

\( = \sum_{i=1}^n \sum_{j=1}^n Cov[X_i, X_j] \)