Poisson Distribution Property

[mathdragon]

Hello, I accidentally stumbled upon this property:

This works for all whole number values for n. This comes from the PMF of a Poisson distribution. Could someone explain to me conceptually why this property works given the properties of a Poisson distribution (i.e. that the Poisson distribution is the distribution with an average amount of time between occurrences)? Hopefully you can understand what I am asking. Thank you!

 

[Harry_the_cat]

This represents the sum of all the probabilities and so must equal 1 if the function is a PMF.

 

[mathdragon]

Thank you for your answer Harry_the_Cat, but that wasn't quite what I was looking for. So, if you look closely this integral is not integrating the pmf from 0 to infinity of possible values of n. Rather, it is integrating all the x-values which should be lambda values which is a parameter. Poisson is a discrete distribution so the sum of the PMF would be a sum, not an integral. This is integrating "sideways" by adding all the possible lambda values for a given number of occurrences, not the other way around. Could someone help me explain why this property is true given this information?