# Random Variable Decomposition Standard Error

I have a decomposed random variable $X$ into partitions $A_1,A_2,\dots A_m$. I know how to compute the expected value of X and the variance of X given the variance and the standard errors of $X$ given each partition:

https://en.wikipedia.org/wiki/Law_of_total_expectation

and

https://en.wikipedia.org/wiki/Law_of_total_variance

What is the variance of the sample mean given $n_1,n_2,\dots,n_m$ samples from each partition $X \vert A_i$. Edit: By this I mean the variance of the sample mean $\tilde{X}:=\sum_{i=1}^m P(X\vert A_i) \frac{\sum_{j=1}^{n_i}x_j^{(i)} \sim X \vert A_i}{n_i}$.

Could it be that it is: $Var(\tilde{X})=\sum_{i=1}^m P(A_i)^2 \frac{Var(X\vert A_i)}{n_i}$? This would assume that $X\vert A_i$ and $X \vert A_j$ are uncorrelated. Are they?

• What does it mean to decompose a random variable into partitions? – dsaxton Jul 2 '15 at 14:24
• This means that the events $A_1,A_2,\dots A_n$ are mutually exclusive and exhaustive. By decomposing, I mean computing statistics on X (mean, variance, standard error) using $X\vert A_i$ instead of X. – Benedikt Bünz Jul 2 '15 at 17:33