If a random variable X follows a poisson distribution, the probability of k events occurring in a given interval of time is given by the following probability mass function:

$f\left(k\right) \= P\left(X\=k\right)\= \frac{{\mathrm{\λ}}^k{\ⅇ}^{-\mathrm{\λ}}}{k\!}$ ,

for $k \= 0\,1\,2\, ..\.$,

Where λ is the constant rate or intensity at which the event occurs and k is the number of events.

The cumulative distribution function is defined as:

$f\left(k\right) \= P\left(X\le k\right) \= {\ⅇ}^{-\mathrm{\lambda }} \sum _{i\=0}^{⌊k⌋}\frac{{\mathrm{\lambda }}^i}{i\!}$,

for $k \= 0\,1\,2\, ..\.$$$

Note that the probability function and cumulative distribution function for the poisson distribution are only defined for integer values for k and as such, there is no continual curve which can be drawn through the point.  The gray lines are only for illustrating the shape of each function over the interval.