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.
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.