Probability calculation of an event.

Question

Suppose we have a village that has the following number of total rain days every year:
A1, A2, A3, ...., An for n years. With Ax an integer number of course.

We want to find the probability BASED on its history, that in the (n+1)th year, this village would have K days of rain (K is an integer number).
Since we speak about weather, we assume the events(number of rain days per year) to be independent.

A random example: Smallville has a history of total rain days per year:
1910: 87
1911: 78
1912: 79
.....
1940: 65
With a mean value for example of 77.5.
Then we want to know:
What is the probability BASED on its history, that Smallville in 1941 would have 75 days of rain?
And what is the probability BASED on its history, that Smallville in 1941 would have between 70 and 75 days of rain?


Should we have to use Poisson distribution and if so how?


Thanks.

Answer 1

Since you are emphasizing that is must be "BASED on history"...you can do no better than the empirical distribution function of your data. Now, just answer your questions from the empirical distribution.

The next level of abstraction would indeed to be to fit a distribution. I don't think that rainy days happen randomly in a given year...you tend to have "fronts", so you'll need to look at the empirical distribution to determine what makes sense.

Answer 2

This is a discrete random variable where a sample $A_i$ is the number of days it rained that year. Natural processes, like days of rain per year are very often fit to a Standard Normal Distribution (and in a stats class you can read this as always).

So use a standard normal lookup table. calculate your $\sigma$ and normalize the bounds with:

$$ B_i = \frac{x-\mu}{\sigma} $$

Where $B_i$ is the normalized bound. Now you want $P\left(\frac{70-\mu}{\sigma}\le Z \le \frac{75-\mu}{\sigma}\right)$. To do this you find your normalized bounds in the table (probably at the front or back of your stats text--or look up online) and subtract the probability that it is less than 70 from the probability that it is less than 75.