Poisson distribution expectation

If we have a poisson distribution, X\sim Po(\lambda), then the expectation of the distribution is just \lambda.

\text{For }X\sim Po(\lambda):\quad E(X) = \lambda

The reason is pretty simple actually. The expectation of a discrete random variable is just the mean value we can expect to get an average of, if we repeat the experiment enough times. And in a poisson distribution, the mean value is \lambda. So the expectation is just \lambda.

Non-discrete expectations

The expectation (mean) does not have to be an obtainable discrete value, or even an integer!

For example, the expectation of the result of a dice roll is 3.5, even though you can never actually roll a 3.5 from a single dice roll. The expectation is just the average value you would get if you rolled the dice many times, and 3.5 is the average of the numbers from 1 to 6.