Law of total variance and its use

ProofIntermediate⏱ 30 minProbabilitytotal varianceconditional expectationdecomposition

Theory: Expectation, Variance and Conditional Expectation →

Problem

Prove the law of total variance

\Var(X) = \E\big[\Var(X\mid\mathcal G)\big] + \Var\big(\E[X\mid\mathcal G]\big),

and use it to compute the variance of $X$ when $N\sim\text{Poisson}(\lambda)$ and, given $N$ , $X$ is the sum of $N$ i.i.d. variables with mean $m$ and variance $s^2$ .

Write a complete argument. Every inequality you use should be one you could state precisely.

Hints

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Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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