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I looked up the definition of Quantile function in Wikipedia, it is said that:

The Quantile function is $Q(p)=\inf\{x\in R:p\le F(x)\}$ for $F:R\to(0,1)$, for a probability $0<p<1$, the quantile function returns the minimum value of x for which the previous probability statement holds.

First, I would like to clarify my understanding of the statement, below is the diagram for $Q(p_3)$, am I correct? Please correct me if I am wrong.

enter image description here

Next, I quite don't understand the use of infimum in the definition, why should we use infimum? Is minimum enough? Could anyone kindly provide an example where using minimum is wrong, or at least does not hold true in general?

Many thanks for any helps.

user71346
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    Good question, I don't understand that either. It seems to me that we could take the minimum since a distribution function is right-continuous. But perhaps we want to cover more general cases, where F is non-decreasing but not necessarily right-continuous (then it isn't a distribution function though)? – kelu Jan 16 '14 at 01:29

1 Answers1

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$Q(p_3)=q_3$ because $F(q_3)>p_3$ whereas $F(q)<p_3$ for any $q < q_3$.

I guess, the problem with $\min$ is that in some cases it might lie outside the domain. For example, when the distribution is uniform over the union of two open intervals $(0,1)$ and $(2,3)$, you might want to consider $(0,1) \cup (2,3)$ as the domain of F. This does not contain $Q(1/2) = 2$.

snw
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