Conditional Expectations and Variances

Here we look at an important concept that is an extension from Bayes Theorem, which we discussed briefly.

The condition expectation identity says \mathbb[X] = \mathbb[\mathbb[X|Y]]
The condition variance identity says Var(X) = Var(\mathbb[X|y]) + \mathbb[Var(X|Y)]

Here both \mathbb{E}[X|Y] and Var(X|Y) are both functions of Y and are therefore random variables themselves.

With this, we start by considering a random sum of random variables. Let W= X_1 + X_2 + \ldots + X_N where X_i‘s are IID with mean \mu_x and variance {\sigma_x}^2, where N is also a random variable, independent of X_i‘s.

\mathbb{E}[W]
= \mathbb{E} [\mathbb{E}[\sum_{i=1}^N x_i | N]]
= \mathbb{E} [N \mu_x]
= \mu_x \mathbb{E}[N]

Var(W)
= Var(\mathbb{E}[W|N]) + \mathbb{E}[Var(W|N)]
= Var(\mu_x N) + \mathbb{E}[N {\sigma_x}^2]
= {\mu_x}^2 Var(N) + {\sigma_x}^2 \mathbb{E}[N]

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