The problem is as follows:
- The population $X$ of 100 samples is drawn from standard normal distribution.
- Then from $X$ 50 samples (let's call them $\widetilde{X}$) are drawn uniformly with replacement.
- The mean of $\widetilde{X}$ is calculated (let's call it $\overline{\widetilde{X}}$).
I need to derive variance of $\overline{\widetilde{X}}$.
I used law of total variance, which states:
$$Var[Y] = E[Var[Y|X]] + Var[E[Y|X]]$$
In my case, this translated to:
$$Var[\overline{\widetilde{X}}] = E[Var[\overline{\widetilde{X}} | X]] + Var[E[\overline{\widetilde{X}} | X]]$$
I derived $Var[\overline{\widetilde{X}} | X]$ and $E[\overline{\widetilde{X}} | X]$ as though $X$ is a known general population, getting new random variables, and then derived their expected value and variance accordingly. The result was the value $$\frac{1}{50} \frac{99}{100} + \frac{1}{100} = 0.0298$$
The question is, am I doing something wrong and is this even legal? Wikipedia states that law of total variance applies only if $X$ and $Y$ are random variables on the same probability space, but that does not seem to hold in this case. But the derived result was consistent with experiments I did, just generating a lot of values following the algorithm above and then calculating their empirical variance.