I really like the Poisson Distribution. A very interesting question I've come across once is:
A given event happens at a rate of every 10 minutes on average. We can see that:
- The expected length of the interval between events is 10 minutes.
- At a random moment in time the expected wait until the next event is 10 minutes.
- At the same moment, the expected time passed since the last event is also 10 minutes.
But then we would expect the interval between two consecutive events to be 10+10 = 20 minutes long. But we know intervals are 10 on average. What happened here?
The key is that by picking a random moment in time, you're more likely to fall into a bigger intervals. By sampling a random point in time the average interval you fall into really is 20 minutes long, but by sampling a random interval it is 10.
Apparently this is called the Waiting Time Paradox.
You went astray when you declared the expected wait and expected passed.
Draw a number line. Mark it at intervals of 10. Uniformly randomly select a point on that line. The expected average wait and passed (ie forward and reverse directions) are both 5, not 10. The range is 0 to 10.
When you randomize the event occurrences but maintain the interval as an average you change the range maximum and the overall distribution across the range but not the expected average values.
When you randomize the event occurences, you create intervals that are shorter and longer than average, so that a random point is more likely to be in a longer interval, so that the expected length of the interval containing a random point is greater than the expected length of a random interval.
To see this, consider just two intervals of length x and 2-x, i.e. 1 on average. A random point is in the first interval x/2 of the time and in the second one the other 1-x/2 of the time, so the expected length of the interval containing a random point is x/2 * x + (1-x/2) * (2-x) = x² - 2x + 2, which is 1 for x = 1 but larger everywhere else, reaching 2 for x = 0 or 2.
I think I understand my mistake. As the variance of the intervals widens the average event interval remains the same but the expected average distances for a sample point change. (For some reason I thought that average distances wouldn't change. I'm not sure why.)
Your example illustrates it nicely. A more intuitive way of illustrating the math might be to suppose 1 event per 10 minutes but they always happen in pairs simultaneously (20 minute gap), or in triplets simultaneously (30 minute gap), or etc.
So effectively the earlier example that I replied to is the birthday paradox, with N people, sampling a day at random, and asking how far from a birthday you expect to be on either side.
If that counts as a paradox then so does the number of upvotes my reply received.
The way, I understand it is that with a Poisson process, at every small moment in time there’s a small chance of the event happening. This leads to on average lambda events occurring during every (larger) unit of time.
But this process has no “memory” so no matter how much time has passed since the last event, the number of events expected during the next unit of time is still lambda.
From last event to this event = 10, from this event to next event = 10, so the time between the first and the third event is 20, where is the surprise in the Waiting Time Paradox?, sure I must be missing some key ingredient here.
The random moment we picked in time is not necessarily an event. The expected time between the event to your left and the one to your right (they're consecutive) is 20 minutes.
I think we must use conditional probability, that is the integral of p(X|A)P(A), for example probability the prior event was 5 minutes ago probabity(the next one is 10 minutes from the previous one (that is 1/2). This is like markov chain, probability of next state depends of current state.
A given event happens at a rate of every 10 minutes on average. We can see that:
- The expected length of the interval between events is 10 minutes.
- At a random moment in time the expected wait until the next event is 10 minutes.
- At the same moment, the expected time passed since the last event is also 10 minutes.
But then we would expect the interval between two consecutive events to be 10+10 = 20 minutes long. But we know intervals are 10 on average. What happened here?
The key is that by picking a random moment in time, you're more likely to fall into a bigger intervals. By sampling a random point in time the average interval you fall into really is 20 minutes long, but by sampling a random interval it is 10.
Apparently this is called the Waiting Time Paradox.