<< Statistics`DiscreteDistributions`

PoissPDF[k_, N_, dt_, f_] := 
      PDF[PoissonDistribution[N*f*dt], k]

?PoissPDF

Trigger[nTh_, N_, dt_, f_] := 
      Evaluate[Sum[
            PoissPDF[k, N, 5/10^9, f]*
              PoissPDF[nTh - k, N, dt, 
                f]*(1 - Sum[PoissPDF[j, N, 5/10^9, f], {j, 0, k}]), {k, 
              0, nTh}]]*(10^9/5)

?Trigger

N[Trigger[133, 81000, 425/10^9, 3 10^3]]

N[%]

N[PoissPDF[249, 81000, 425/10^9, 6 10^3]*81000*6*10^3]