This is my result from finishing a problem in a stochastic process workbook:
"Since a Poisson process has the property of stationary increments, this implies that the inter-arrival times $Z_n=T_n-T_{n-1}$ are identically distributed without regard to the starting value of n. That is:
$$\Pr(Z_{n-1} \le z) = \Pr(Z_n \le z)$$
$$F_{Z_{n-1}}(z) = F_{Z_{n}}(z)$$
For inter-arrival times of a Poisson Process, the CDF of step n-1 equals CDF for step n
How does this property relate to the memoryless property for exponential random variable, namely:
A RV X is memoryless if X is positive, and for all real $t>0$ and $s >0$, then:
$P[X > t +s] = P[X>t]~P[X>s]$
A RV X is memoryless if for any s and t, we have:
$P(X > t+s~|~X > t) = P(X>s)$
I was just curious... because CDF and memoryless properties seem similar to each other ... but the sign is backwards...