8
$\begingroup$

I understand that inter-arrival times of a Poisson process are exponentially distributed and therefore the inter-arrival times are memoryless.

However, how about the waiting times of Poisson process i.e wait time till $k$th arrival where $k \geq 2$. This is an Erlang distribution and shouldn't be memoryless -- or is it?

If it is, can someone help how to show it?

In general, why is Poisson process memoryless? I understand that interarrival times or time to next arrival are but doesn't look like time till kth arrival is also memoryless .. or is it?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
10
$\begingroup$

Memorylessness is a property of the following form:

$$\Pr(X>m+n \mid X > m)=\Pr(X>n)\ .$$

This property holds for $X_1=\ \text{time to the next event in a Poisson process}\ $, but it doesn't hold for $X_k=\ \text{time to the}\, k^\text{th}\, \text{event in a Poisson process}\ $ when $k>1$.

As for how to show it, you could try to do it from first principles.

If you can show that the essentially equivalent form $P(X>s+t)\neq P(X>s)P(X>t)$, (for $s, t>0\ $), that would be sufficient; you already know the distribution for $X_k$.

Improve this answer
$\endgroup$
4
  • $\begingroup$ Thanks. You address confusion that i had.So why is poisson process called memoryless when its only X1 thats memoryless and this is not true for Xk with k>1? $\endgroup$
    – toing
    Oct 7, 2014 at 1:30
  • $\begingroup$ As an extension, let me also ask if poisson distribution( not poisson process ) is memoryless? $\endgroup$
    – toing
    Oct 7, 2014 at 4:54
  • 2
    $\begingroup$ What is memoryless in the Poisson process is inter-event intervals (usually inter-event times). The Poisson distribution itself is the distribution of counts per unit interval. So you have a set of counts, but not the times (or whatever you're measuring events over). You don't have inter-event intervals to be memoryless about. $\endgroup$
    – Glen_b
    Oct 7, 2014 at 5:03
  • 1
    $\begingroup$ You can say $\Pr(X_k>m+n \mid X_1 > m)=\Pr(X_k>n)$ which is not the memoryless property but is related to it $\endgroup$
    – Henry
    Aug 1 at 22:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.