Conditional Expectations and Variances

By KS Teng

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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