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.

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

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

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for the more than 10 years. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

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