Mean of a Poisson with an exponential rate

Question

The rate $\lambda$ is drawn from an exponential distribution with mean $1$ , and given $\lambda$ , $X$ is Poisson with mean $\lambda$ . What is $\mathbb{E}[X]$ ?

Short answer

$1$ . By the tower rule, $\mathbb{E}X=\mathbb{E}[\mathbb{E}(X\mid\lambda)]=\mathbb{E}\lambda=1$ .

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Lessons behind this question

Conditional Expectation and Variance Revisited
Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating

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