Why is Poisson distribution's expected value not on the curve's maximum

Question

For a Poisson distribution of $Poisson(\lambda=5)$ I have a curve like this:

If the expected value for a Poisson distribution is its rate parameter $\lambda$, then why isn't the maximum on the curve also 5? Doesn't expected value mean the most likely value?

Answer 1

To have mean = mode you need a symmetric distribution. Try with a poisson with $\lambda =10$ or more and you will have $\text{argmax}(x) =\lambda$

Answer 2

This phenomenon is not particular to a Poisson distribution. It's true whenever the distribution is skewed.

Which of the three "averages" you use in any particular application depends on the context.