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I was going through Poisson distribution and I understand the other assumptions made in Poisson distribution except for the last one which is:

The probability of an event in a small sub-interval is proportional to the length of the sub-interval.

Or

The actual probability distribution is given by a binomial distribution and the number of trials is sufficiently bigger than the number of successes

On the same page, there is an example of goals in a soccer match which follows the Poisson distribution.

Let us assume that there are 64 matches in a soccer world cup. It is given that $\lambda=2.5$. Let us define an event as '5 goals scored in a match' and consider two sub intervals, one which consists of four semifinal matches and so its length is four and another sub interval which consists of eight quarter-final matches and so its length is eight.

Since this random variable follows Poisson distribution and so will satisfy all the assumptions made in Poisson distribution.

My questions are:

1. What is the meaning of 'probability of a certain event in any sub-interval'?

In our case, it will become: probability of scoring 5 goals in semi-finals(/ quarter-finals)

2. What is the meaning of proportionality when we say that 'probability of an event in a small sub-interval is proportional to the length of the sub-interval'?

3. Why is it written 'small' sub-intervals?

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Since a Possion distribution models 'objects' arriving randomly and independently in time or, space, the probability of that event occurring in any sub-interval given you know it has occurred already, is uniform. I believe that's where the proportionality part comes in.

Let me give you another scenario.

Suppose a grocery store opens at 8:00am daily, and customers arrive to the store according to a Poisson process. Mike, the shift manager, sees a customer in aisle three at 8:03am and says to himself, "I wonder what the probability is that they entered the store in the last three minutes?"

To answer Mike's question for him recall, "the probability of that event occurring in any sub-interval given you know it has occurred already, is uniform."

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I'm new to statistics and I know its kinda late but I'll try to answer in case its helpful for someone. I THINK its referring to the assumptions you make when you derive the Poisson distribution. So I'd recommend looking up how to derive the Poisson distribution. The textbook: mathematical statistics with applications 7th edition is pretty good. But I'll try to give a poorly constructed brief explanation here haha. So.. the way you derive the Poisson distribution is by modeling it after the Binomial distribution. Therefore it needs to satisfy the following conditions (to be a Binomial distribution):

  1. Consists of a fixed number (n) of identical trials
  2. Each trial has 2 outcomes
  3. P(success) = p = same for every trial
  4. Trials = independent
  5. Random variable Y = # of successes out of n trials

So what they did to get to a Poisson distribution was said given some interval, lets split the given interval (time period/space) into n sub-intervals such that each sub-interval is only big enough for an event to occur once. So that way each sub-interval (trial) has only 2 outcomes: either an event happens or doesn't. They then claim that probability of an event occurring in a sub-interval is p for every trial, and the occurrence of an event is independent sub-interval to sub-interval.

They then define lambda as lambda = n*p, and then say hey lets make a ton of sub-intervals (n) relative to p and everything. So they then take the limit as n goes to infinite of the probability distribution of the Binomial distribution (nCy * p^y * (1-p)^(n-y)). They then substitute p = lambda/n and eventually get to the Poisson distribution and they're done.

So to answer your questions:

  1. Its just the probability of an event happening in one of those sub-intervals previously mentioned.

  2. So.. uhh as the number of sub-intervals (n) increase, the length of each sub-interval decreases (cause you're splitting your fixed time period into more parts), and therefore because the length of a sub-interval decreases, the chances of an event occurring in these shorter sub-intervals also decreases proportionately. Like idk.. if you're walking down the street for an hour, you see a person on average once every 15min usually, your initial interval is the hour, but as you split it more and more (n number of sub-intervals increases) the chances of you seeing someone decreases. Like if you split it into 60 minutes or 60 sub-intervals then you're less likely to see someone in a minute compared to seeing someone in an hour. The reason this is important is because when you're calculating the limit as n approaches infinite of the binomial distribution, if p didn't decrease proportionately as n increased then lambda = n*p would just hit infinite or something and your final probability distribution wouldn't be very useful probably.

  3. The sub-intervals have to be small because remember you're modeling the Binomial distribution and your sub-intervals are practically your trials. Also recall that a Binomial distribution only has 2 outcomes. Your sub-intervals are defined to be small enough that only one of two things happen: either an event occurs once or it does not occur at all. If your sub-intervals were not small, then it becomes possible that more than 2 things could happen: no events occur, one event occurs, two events occur, etc. So yeah they're small cause they're modeling the Binomial distribution.

I hope that helps.. if someone who is more knowledgeable about statistics and the Poisson distribution can confirm what I said is correct, that'd be great. Someone should also edit the wikipedia page to explain cause if you didn't know how the Poisson distribution was derived then I have no idea how you could understand why the assumptions are made.

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