Expectation of Poisson Distribution

Theorem
Let $$X$$ be a discrete random variable with the Poisson distribution with parameter $\lambda$.

Then the expectation of $$X$$ is given by:
 * $$E \left({X}\right) = \lambda$$

Proof
From the definition of expectation:
 * $$E \left({X}\right) = \sum_{x \in \operatorname{Im} \left({X}\right)} x \Pr \left({X = x}\right)$$

By definition of Poisson distribution:
 * $$E \left({X}\right) = \sum_{k \ge 0} k \frac 1 {k!} \lambda^k e^{-\lambda}$$

Then:

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