Tail of probability distribution

Question

I need to analyze the plot of a probability distribution for a group of random samples. The question asked: "What does the tail of probability distribution of the sample values look like?" I don't know how should I answer this question. Are there specific categories and definition for the tail of probability distribution?

Can anyone provide an insight (or introduce a source) please?

Thanks in advance.

Answer 1

This essentially means how much probability is distributed over the largest values(usually) of the random variable. A distribution may be viewed as light tailed- if it assigns smaller probabilities for larger values of the variable, or heavy tailed - if it assigns larger probabilities for larger values of the variable.

Q-Q plots are usually employed to detect the presence or absence of heavy tails.

Tail weight may help use choose a model in a given situation. If we have two models to choose from to model a characteristic and we want to go for a heavy tail distribution, then we may consider limit of either the ratio of two probability densities or the ratio of the survival functions associated with those densities. If these ratios diverge to infinity, the random variable associated with the numerator is said to have heavy tail. Here tail weight is a relative concept.

A characteristic of light tail distributions is that all positive moments for the distribution exist while for heavy tail distributions they exist only upto a certain value.

An analysis of hazard rate function also gives some information about distribution tails. An increasing hazard rate indicates a lighter tail and a decreasing hazard rate means a heavy tail.

The central t-distribution will have heavy tails when the degrees of freedom is small and lighter tails for large degrees of freedom.

Gamma and lognormal distributions come under light tail distributions and Pareto distribution is an example of heavy tail distribution.