Note that a Poisson random variable will ALWAYS be a non-negative number. It CANNOT have mass on the negative end of the real line. So the plot that you show, with a supposedly Poisson random variable that goes negative? It cannot exist. (At least not for a standard Poisson.)
Perhaps the intent is to have a Poisson random variable that has been transformed into the range of a normal distribution.
For example, Poisson distributions with large Poisson parameters will tend to look very normally distributed. (Not difficult to prove as I recall.)
So a Poisson CDF that does look quite normal. A quick glance at Wikipedia...
tells me that for a Poisson distribution with parameter lamnda, the mean will be lambda, as well as the variance. So we can simply do this:
s you should see, the two curves nearly overlay on top of each other.
In your figure, the Poisson was apparently transformed, via a transformation to look like a normal.
fplot(@(x) poisscdf(lambda + x*sqrt(lambda),lambda),[-3,3])
However, that is NOT a Poisson distibution because it is shown to have mass for negative x. It is derived from one.