What does stationary independent increment mean here?

2 votes

Provide short answers to the following questions:

$P_{n} (t)$ is the probability of $n$ events occurring during a time interval $t$. How will you express $P_{0} (t + h)$ in terms of $P_{0} (h)$, if $P_{0} (t)$ has stationary independent increments? (Note: $P_{t} (t)$is the probability density function).

$P_{n} (t)$ is the probability of $n$ events occurring during a time interval $t$. How will you express $P_{0} (t + h)$ in terms of $P_{0} (h)$, if $P_{0} (t)$ has stationary independent increments? (Note: $P_{t} (t)$is the probability density function).

1 vote

Best answer

$P_0(t)$ denote the probability that no events happened in an interval of length $t.$$$P_0(t + h) = P_0(t) P_0(h)$$ This is because if there are no events in interval $[0,t+h]$ then there are no events in intervals

- $[0,t]$
- $[t, t+h]$

These two intervals are non overlapping and it is given in question that $P_0(t)$ has stationary independent increments and so their probabilities are independent.

PS: One of the axioms of Poisson distribution is that the numbers of events in two nonoverlapping regions are independent.