Property 1
An exponential distribution is memoryless
Proof: We show that P(x ≥ s) = P(x ≥ t + s|x ≥ t)
Property 2
If x has a Poisson distribution with mean λ, then the time between events follows an exponential distribution with mean 1/λ.
Proof: if we have a Poisson process with mean λ, then the mean number of events that occur in time t is λt. Thus the probability that no events occur in time t is
where x = the number of events that occur in time t.
Now let y = the time until the first event. Then from the above, we see that
and so
which is the cumulative exponential distribution of y.
Reference
Wikipedia (2012) Exponential distribution
https://en.wikipedia.org/wiki/Exponential_distribution