μ=.;
pdf=PDF[PoissonDistribution[μ],x]

poisDist[μ_,x_]=pdf[[1,1,1]]

Plot the Poisson Distribution Function for a mean value of 3.2

Notice that the value of x does not have to be an integer.   
“Smart Mathematica” uses the Γ function to evaluate non-integer factorials.

μ=3.0;
a=1.;
Plot[a poisDist[μ,x],{x,0,10}]
μ=.;

Take notice of a couple of things:

1.  The area underneath the Poisson function is normalized, independent of the mean value μ,

PDF[PoissonDistribution[μ],x]

CDF[PoissonDistribution[μ],∞]

2.  The mean value of the Poisson function is μ.

Mean[PoissonDistribution[μ]]

3.  The variance (i.e.,  (σ^2) ) is μ .     So, the standard deviation   σ = (Sqrt[μ])

Variance[PoissonDistribution[μ]]