Expectation and Variance of Poisson Distribution equal its Parameter

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

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 4.3$: Moments: Exercise $5$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Poisson distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Poisson distribution