Consider the following random variables in $(\Omega, \mathfrak{F}, P)$.
- $X_1,X_2, X_3,...$ where $\forall n \in \mathbb{N}, \mu_n = E(X_n), \sigma_n^2 = Var(X_n) < \infty$.
- $N$, a discrete RV whose range is $\mathbb{N}$. Let $A_n = ({\omega: N(\omega) = n})$ and $p_n = P (N = n) = P(A_n)$. Also, $N$ is independent of each $X_n$.
- $\mu_N$ defined as $\mu_N (\omega) = \mu_{N(w)}, \sigma_N^2 (\omega) = \sigma^2_{N (\omega)}$, and $X_N$ defined as $X_N (\omega) = X_{N(\omega)} (\omega)$.
Interpretation: if we choose an index n at random (so $N=n$), then $X_N$ is the RV $X_n$ and $\mu_N$ is the mean of $X_n$, and $\sigma_N^2$ is the variance of $X_n$.
Prove that $E[X_N|N] = E[X_N|\sigma(N)] = \mu_N$ and $Var[X_N|N] = Var[X_N|\sigma(N)] = \sigma_N^2.$ Use these to find formulas for $E[X_N]$ and $Var[X_N]$ in terms of $\mu_n, \sigma^2_n$ and $p_n$.
My attempt:
Edit: I already proved the conditional expectation stuff. I just want to see if my formulas for $E[X_N]$ and $Var[X_N]$ in terms of $\mu_n, \sigma^2_n$ and $p_n$ are correct:
$Var[X_N] = E[Var[X_N|N]] + Var[E[X_N|N]]$
$=E[\sigma_N^2] + Var[\mu_N]$
$=E[\sigma_N^2] + E[\mu_N^2] - E[\mu_N]^2$
$=\sum_{n=1}^{\infty} \sigma_n^2 p_n + \sum_{n=1}^{\infty} \mu_n^2 p_n - (\sum_{n=1}^{\infty} \mu_n p_n)^2$
Explanation of sums and stuff: (For $E[\mu_N]$)
$E[X_N|N]$ is $\sigma(N)$-measurable. Thus, $\exists$ Borel-measurable function g such that $g(N) = E[X_N|N]$
$\to E[E[X_N|N]] = E[g(N)]$
$\to E[X_N] = E[g(N)]$
$\to E[X_N] = \sum_{n=1}^{\infty} g(n) p_n$
where $g(n) = E(X_N | N = n) = E[X_n 1_{A_n}]/p_n = E[X_n]p_n/p_n = E(X_n) = \mu_n$
$E[X_N] = E[\mu_N] = \sum_{n=1}^{\infty} \mu_n p_n$
Similarly, I came up with expressions for $E[X_N^2]$ and $Var[X_N]$.
Are these expressions correct?