Expectation and Variance of Poisson Distribution equal its Parameter

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Theorem

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


Then the expectation of $X$ equals the variance of $X$, that is, $\lambda$ itself.


Proof

From Expectation of Poisson Distribution:

$\expect X = \lambda$

From Variance of Poisson Distribution:

$\var X = \lambda$

$\blacksquare$


Sources