Forgive my ignorance, I am brand new to Poisson and statistics in general.
$$
\bbox[5px,background:black]{\color{white}{\begin{array}{l}
\text{Poisson Distribution}\\
P(X=k)=\frac{\lambda^ke^{-\lambda}}{k!}\\
k\text{ is the given number of event occurrences}\\
\lambda\text{ is the average rate of event occurrences}
\end{array}}}
$$
original image
My statistics class has this formula for figuring out the probability of a less-common event happening given the average rate of occurance.
The example is "An intersection has, on average, 15.5 accidents per week. Using the Poisson Distribution formula, determine the probability of the intersection having only 1 accident in a week."
Ok, easy, just plug and chug and you get your answer.
Here is my actual question: How could a formula possible answer this? What if the average was 15.5 but always ranges from 10 to 20 per week? What if 100 years went by and every week there was somewhere between 10 and 20 accidents, givin us an an average of 15.5 but NEVER having just a single accident? Or what if the average was 15.5 because it is always either 15 or 16 per week? The point being, there may be a near-zero probability of 1 or 14 (or any value of 'k') happening, yet the formula just guesses its little heart out having no clue that it's basically a 0% chance (and yes, I know that probabilities are all based on information we have and don't have, I get that, just a way for us to make a best guess using the information we have.)
This is the most simple example I can come up with to explain why I am confused about how this could work. It seems like to get anything even slightly meaningful you would have to have at least some information regarding the max and min or something to give you at least some sort of range, right?
Thank you in advanced! I am decent at math but I am not a mathematician or anything like that. Just a CS student with a lot more math to learn haha. But I am very curious, and this is bothering me.
Best Answer
Good question! We need to make a modeling assumption here, namely that accidents occur independently of each other; that is, we assume that in some small interval of time $\Delta t$ (say an hour, for the purposes of this problem) there's some small probability $p$ that an accident will happen, and whether an accident occurs in any given small interval is independent of any other.
With this assumption (which is very strong), the number of accidents that occur in a large interval of time is a binomial distribution. Specifically, if we fix the average number $\lambda$ of accidents and divide the week into $N$ small time intervals where the probability that an accident occurs is $\frac{\lambda}{N}$ (this is necessary for the average to be $\lambda$), then the distribution of accidents is roughly $\text{Bin}\left( N, \frac{\lambda}{N} \right)$.
Now the interesting mathematical fact is that as $N \to \infty$ this binomial distribution converges (quite rapidly) to a Poisson distribution $\text{Pois}(\lambda)$; this is the simplest special case of the Poisson limit theorem. In terms of the probability that there will be $k$ accidents this says that
$$\lim_{N \to \infty} {N \choose k} \left( \frac{\lambda}{N} \right)^k \left( 1 - \frac{\lambda}{N} \right)^{N-k} = \frac{\lambda^k}{k!} e^{-\lambda}$$
which you can check by writing ${N \choose k} = \frac{N(N-1) \dots (N-k+1)}{k!}$ and using the fact that $\lim_{N \to \infty} \left( 1 + \frac{x}{N} \right)^N = e^x$. But this limit calculation doesn't really show you how fast the convergence is; you can check that things are already quite close for, say, $N = 168$ (the number of hours in a week).