def pdf_direct(self, tt, N=50):
a = self.K
t = tt/self.tau
S = np.ones_like(t)
s = np.ones_like(t)
for k in range(1, N):
s *= (a*t)/(k*(k+1.))
S += s
return np.exp(-t)*a/np.expm1(a) * S /self.tau