# When to use which random variable?

Given a population ($\Omega,A,P$) and a random variable $X : \Omega \to \mathbb{R}$ defined on it, we want to predict the distribution of $X$ intuitively.Once we know that $X$ follows a particular distributin we can calculate many things like the moments (if they exist) and quantile. Depending on what the random variable is measuring we learnt intuitively find the distribtion for some cases like the following ( in our class ):

1) Measuring success or failure of an element in the population $\to$ Bernoulli.

2) Counting the number of successes $\to$ Binomial.

3) Waiting time for one success $\to$ Geometric.

4) Waiting time for $n$ successes $\to$ Negative Binomial.

5) counts of events in a global scale which are locally rare $\to$ Poisson.

6) Given a sample containing two types of elements, counting the number of one type $\to$ Hypergeometric.

I want to know how this can be extended to the case of continuous random variables like the following:

1) ? $\to$ Uniform

2) ? $\to$ Exponential

3) ? $\to$ Gamma

4) ? $\to$ Normal

If there are any good references for any of these and also for other random variables not mentioned here , please provide the links for them.

predict the distribution of X intuitively

this is the problem of fitting a distribution to data, which consists of finding a mathematical function which represents in a good way a statistical variable. This is usually done by experience:

1. when you have a set of data, based on its clinical meaning and on the histogram of observed data, you can define a family of distribution that your data can eventually fit.
2. From the family of distribution that you define previously, you can estimate the parameter(s) for each single distribution.
3. With the estimated parameters, you can evaluate the goodness of fit following some methodologies/criteria.

Sometime, we face a situation that there is no absolute distribution that fit best to the data, may be there are 2-3 distributions with more or less the same quality of fit, may be your data is the average/sum of two/three distributions.

• So,you are saying that we come up with a few guesses ( based on the sample we collect from population ) and choose the best fit. You said that it " is usually done by experience " which is what I want to know about. From your experience, can you share when you come across normal, uniform, gamma and exponential ? Sep 15, 2015 at 16:03
• Very good question, the choice of the family of distribution is based not only on the personal experience, but also on the companies' experience as well. I don't know for other industry, but I'm working on the insurance industry. I will take a simple example: The claims paid by an insurance companies is: $$S=\sum\limits_{i=1}^N X_i$$ where: N is the (random) number of claims in a year $X_i$ is the amount of money paid for each claim Sep 15, 2015 at 16:25
• People ussually come up with several kinds of distribution for N such as: Poisson, Binomial, Negative Binomial, ... And for $X_i$: Exponential, Gamma, Normal, Log-normal, Weibull, Pareton ... or even the mixed distribution between Exponential and Erlang, ... Of course you can follow the traditional way to fit a distribution, or follow your experience, and finally, you say, OK, N follows the Poisson distribution, $X_i$ follows the normal distribution. But in pratice, each company has their own experience and end up with a different set of "possible distribution" than yours. Sep 15, 2015 at 16:36
• And perhaps, based on the companies' experience, they will end up with the different fitted distribution than yours. And usually, it is very hard to conclude between theirs and yours, which is the better one. Sep 15, 2015 at 16:38