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I'm trying to understand Poisson Distribution but I can't find a good explanation for one of the sentences above. I've highlighted it within a red rectangle - "The probability of an event in a small interval is proportional to the length of the interval". What does that mean?

Thanks.

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    If $I=[0,1]$, when $n$ is sufficiently large, we have $I_i=[(i-1)/n,i/n]$. In each $I_i$, there is only one event occur and the probability is $k/n$. – 2017-01-22
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    @Simple hi, sorry, what is I? is I a probability? – 2017-01-22
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    $I$ is an interval. – 2017-01-22
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    I see. In that case, why do we have "n" within the interval of I(i). Without the "n", I would interpret I(i) as the interval between event i-1 and event i. I can't understand what's the purpose of the "n" in the interval. – 2017-01-22
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    If we have a sufficiently large $n$, in each $I_i$ is a small interval and there is **only one** event in $I_i$. The purpose of $n$ is to make sure there is only one event occur in a small interval $I_i$. – 2017-01-22
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    Sorry, I'm confused with what "k" is. Is "k" the number of events occurring within "n"? And is "n" the total number of trials within the entire experiment or is n the number of trials within an interval I(i)? – 2017-01-22
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    yes, $n$ is the total number of trial. $k$ is a poisson random variable, in general, I use $\lambda$. – 2017-01-22

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Imagine you flip a very biased coin such that the probability of heads is $1/60$. And say you flip it once a second. Then, you'd get on average one heads per minute. But what is the full distribution of the number of heads you get within a minute?

The Poisson distribution is the approximate answer to this question. The number of heads you get within a minute is approximately Poisson distributed with mean 1. This approximation would be even better if we flipped a coin with $1/6000$ probability of heads every hundredth of a second for a minute.

In general a poisson distribution idealizes the situation where you have a large number of independent trials $n$ and a small probability of success $p$ such that $pn$ is order 1. In other words, if you make $n$ very large and set $p = \lambda /n$ then the distribution of number of successes in the $n$ trials is Poisson($\lambda$).

To make sense of the highlighted passage, what they are trying to get across is the notion I expressed in my first example where we improved the approximation by flipping the coin with $1/6000$ probability every hundredth of a second. Notice that we're flipping coins at such a rate that they kind of blur together. The trials are going on continuously with very small probability of individual success, but there is a rate of success of 1 success per minute.

So if you look in a small time interval compared to the rate but large enough that it contains many individual trials - say one second for this example - the probability of having a success will be about $1/60.$ If you looked in a time interval of half a second, it would be about $1/120,$ for two seconds about $1/30,$ etc In this sense for small time intervals, the probability of success approximately proportional to the length of the time interval. The proportionality constant is the success rate.

When the time interval gets comparable to the rate this breaks down since there is a significant probability of having two, three, four successes.

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    hi there! your first paragraph says "full distribution", what does that mean? – 2017-01-22
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    1 success per minute just tells you what happens on average. It tells you nothing about the fluctuations about that average. For instance you could always get exactly one success, or you could get zero sometimes and two sometimes but it averages out to one. The "full distribution" gives you the probability that there's zero successes, the probability that there's one success, the probability that there's two successes, etc. For instance the poisson distribution says that the probability that there are n successes is $\lambda^ne^{-\lambda}/n!$ where $\lambda$ is the average number of successes. – 2017-01-22
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    Oh! Does this mean that we are basically using the average (mean) as the starting point to find all possible outcomes, and the collection of outcomes is known as the "distribution"? – 2017-01-22
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    @glendon Consider it like an experiment where we flip a biased coin 60 times like in my example. The events we're interested in is how many heads we get, which could be anything from 0 to 60. The 'distribution' for these events gives a probability for each possibility 0 to 60. In this case cause the coin only has a 1/60 chance of coming up heads, the probability of 1 heads would be pretty high, as would zero and two, but the probability of getting more than 5 or 6 or so would be pretty small, getting 60 would be astronomically tiny. The values of these probabilities 0-60 is the distribution. – 2017-01-22
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    thanks! great explanation! now i'll try to understand what Simple is trying to teach me. :) – 2017-01-22