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.
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?