Hi, Can someone please help clear something up. If X~Poi(5) distn where X is the number of goals scored in a football match, are we saying the Poisson event is a single goal or the match itself (I'll assume the former for my next question)? Also since the time between Poisson events is exponentially distributed, are we saying that the time between each goal diminishes as the match goes on? Thanks, Stewart.
For a poisson process, events occur at random, independently of the past, but with some long term average of lambda per unit time (in your case, 5 per football match, and the event is a goal). The time between each goal can be approximated by an exponential process, but you should remember that the exponential process is memoryless. In other words, it doesn't matter how long there has been since a goal occurred (whether it just happened last second or 89 minutes ago), the expected time until the next goal is the same. So the expected time between goals does not diminish over time.
Right - so the exponential property is more that the longer we wait between goals, the more likely we are to get a goal (I guess this is true for any dist) but the expectation grows with exponential speed (although thats probably not the right way to word it).
Not exactly. Each goal is a Poisson event, and each one restarts the "exponential clock", so the time between each goal has the same distribution. The point about the memoryless property is that you can't make any inference about the waiting time until the next goal from the time elapsed since the last goal. If you get a goal on average every ten minutes, the fact you have waited twenty minutes without a goal doesn't make a goal in the next few minutes more likely.
Ok - that all makes sense. I am just trying to picture the exponential nature of the waiting time but maybe I'm looking at it the wrong way. Anyway your explanations have been very helpful - thanks.
It's a confusing business. I found the stuff in CT6 on Poisson processes very useful in consolidating my understanding of this area.
