Hey,

1.)

I have a problem. I have to solve 50% quantile of The Poisson Distribution with the middle value of 2.

2.)

Is the factor limited only to selected allowable values? Yes or No?

The Poisson is a discrete distribution whose domain is the set of positive integers. I don't think because of the e it is possible for there to be an integer m such that P(X <= m) = .5 exactly. You could find a range for two integers m and m+1 so that the median .5 falls between P(X <= m) and P(X <= m + 1) .

Wikipedia says the median will lie between lamda - ln 2 and lamda + 1/3.