i require a formula to calculate the standard deviation using variances of three or more variables (lets call them a,b,c) and the covariances between them. To complicate matters more i only need a percentage of all three totalling 100%, so for example a = 50%, b = 40% and c = 10%. Can anybody point me to the right direction as to how i can accomplish this please?
thanks in advance
Best Answer
It looks as if you want the variance of $r X+s Y+tZ$, where $X$, $Y$, and $Z$ are random variables, and $r$, $s$, and $t$ are constants. (It so happens in your problem that $r+s+t=1$, but we will not be using that fact.) The required formula is $$\text{Var}(rX+sY+tZ)=r^2\text{Var}(X)+s^2\text{Var}(Y)+t^2\text{Var}(Z)+2rs\text{Cov}(X,Y)+2st\text{Cov}(Y,Z)+2tr\text{Cov}(Z,X).$$
For the standard deviation of $rX+sY+tZ$, take the square root of $\text{Var}(rX+sY+tZ)$ computed by the above formula.
Remark: The above formula can be derived from the definition of variance. The calculation is algebraically straightforward, but not particularly enlightening. I can write out some details if that would be helpful.
The formula can also be derived from the two-variable version of the formula, which may be familiar to you: $\text{Var}(cU+dV)=c^2\text{Var}(U)+d^2\text{Var}(V)+2cd\text{Cov}(U,V)$.