I'm writing something involving random variables and I'm not sure about the notation being right or not, so I'd really appreciate it if someone could take a quick look and confirm.
We have a sample of $n$ iid discrete random variables $X_1, X_2, ..., X_n$, and then a function $f$ that maps them onto another sample of variables $f(X_1), f(X_2), ..., f(X_n)$. The true mean of this distribution is:
$\mu_{f(X)}=\sum{f(x)P(X=x)}$
Because we only have a sample, we use the sample mean as estimator. The expectation is:
$E[\hat\mu_{f(X)}]=\frac{1}{n}\sum{f(X_i)}$
However, the true mapping function $f$ is unknown, we only have an empirically estimated $\hat f$ (cubic) function. This means that we actually have $\hat f(X_1), \hat f(X_2), ..., \hat f(X_n)$, so the estimate above is:
$E[\hat\mu_{\hat f(X)}]=\frac{1}{n}\sum{\hat f(X_i)}$
$Var[\hat\mu_{\hat f(X)}]=\frac{sd_{\hat f(X)}^2}{n}$
I wonder if there are too many hats in there. Thanks in advance.