In Poisson distribution, the probability of inter arrival time to be t or less is:
$$ P(X\leq t)= 1 - P(X>t) = 1 - P(0 \mbox{ arrivals in } t) = 1 - e^{-\lambda t} $$
and probability of one arrival in t is:
$$ P(k=1)= \lambda t e^{- \lambda t} $$
I wonder how the exponential distribution can be derived from Poisson to reach:
$$ P(X\leq t)= \lambda e^{- \lambda t} $$
Thanks,