StatsPoissonCDF

I'm having difficulty using StatsPoissonCDF, where the values don't seem quite right to me. When I generate the pdf function using StatsPoissonPDF, everything seems as it should be. For example,

StatsPoissonPDF (x = 0, lambda = 1) returns 0.368....
StatsPoissonPDF (x = 1, lambda = 1) returns 0.368....
StatsPoissonPDF (x = 2, lambda = 1) returns 0.184....

Those match theory. If I put those in a wave and integrate, all looks fine.
But, when I use StatsPoissonCDF,

StatsPoissonCDF (x = 0, lambda = 1) returns 0.736...
StatsPoissonCDF (x = 1, lambda = 1) returns 0.736...
StatsPoissonCDF (x = 2, lambda = 1) returns 0.920...

I am okay with CDF (1, 1) and CDF (2, 1), but I don't get CDF (0, 1). Shouldn't that be 0.368...? That's what the integral shows. I am getting similar behavior regardless of the choice of lambda. Did I miss something here? Is this a rounding error? I'm not sure how to get the true 0 value. I'd be grateful for a suggestion.

-Matthew