When to use which random variable?

Question

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.

Thanks in advance.

Answer 1

This is not about

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.