This lecture introduces and discusses the notion of quantile of the probability distribution of a random variable. We start by giving a formal definition of quantile and we then discuss its meaning.
In order to understand the definition of quantile, you need to remember how the distribution function of a random variable is defined:
Quantiles are defined as follows.
Definition Let be a random variable having distribution function . Let . The -quantile of , denoted by is
When the distribution function is continuous and strictly increasing on , then the smallest that satisfiesis the unique that satisfies
Furthermore, the distribution function has an inverse function and we can write
Example If a random variable has a standardized Cauchy distribution, then it distribution function iswhich is a continuous and strictly increasing function. The -quantile of is
However, in many cases the distribution function is not continuous or not invertible (or both) and we need to apply the definition above in order to derive the quantiles of the random variable. The following example illustrates one such case.
Example Let be a discrete random variable with supportand probability mass functionThe distribution function of iswhich is clearly not invertible. Now, suppose we want to compute the -quantile for . There is no such thatHowever, the smallest such that is because for and for . Thus, we have
When there is an such that , the quantile can be interpreted as a cut-off point: with probability , the realization of the random variable will be less than or equal to ; with probability , it will be greater than .
When is regarded as a function of , that is, , it is called quantile function.
The quantile function is often denoted by
As we have already shown above, when the distribution function is continuous and strictly increasing on , then the quantile function coincides with the inverse of the distribution function.
More details about quantiles can be found in the following subsections.
Some quantiles have special names:
if , then the quantile is called median;
if (for ), then the quantiles are called quartiles ( is the first quartile, is the second quartile and is the third quartile);
if (for ), then the quantiles are called deciles ( is the first decile, is the second decile and so on);
if (for ), then the quantiles are called percentiles ( is the first percentile, is the second percentile and so on).
Although the definition of quantile given above is the one that is usually employed in probability theory and mathematical statistics, there are also other slightly different definitions that can be given. For a review, see http://mathworld.wolfram.com/Quantile.html.
Note that in the definition of quantile, we have imposed . This is because for , we have that whatever the distribution of . Instead, for , we have that is not well-defined in general: for example, if has a normal distribution, thenis not well-defined because the setis empty.
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