Hi everyone,
I am trying to derive the variance of a Random Variable as part of my research.
I wanted to check if I have simplified the expression in the word document attached correctly. In other words, if I take the variance of a sum or difference of linear combinations (series) of random variables, is that equal to the sums of the variances of the random variables, where the different series differ by a constant "alpha" (see the document). Note that the random variable in each series is the same random variable.
I tried deducing this by using a few properties of variance:
1. Var(aX) = a^2 Var(X)
2. Var([series from 1 to n of] X) = n Var(X)
3. Var(aX + bY) = a^2Var(X) + b^2Var(Y), where X and Y are independent RVs.
However, I am not sure if this correct. Please could you check if my understanding of this is correct.
Thank you.
If X_1,...,X_N are independent random variables, then your derivation is correct.
For 3. Assume X and Y have zero mean (no change, just easier expressions)
Var(aX+bY)= E((aX+bY)^2) = aVar(X) + bVar(Y), because, since X and Y are independent hence E(XY)=0
Your derivation is correct if the random variables are independent of each other.