2.4 Quantile Functions
A quantile of the cdf \(F_X\) is a value that divides the area under the density into two parts of specified size. Only the area to the left need be named, since the area to the right is what is left over.
Definition 2.21 (Quantile). For \(0<P<1\), the \(P^{th}\) quantile of \(X\) is a value \(x_P\) such that \[P\left (X\leq x_P\right )=P, \hspace {0.8cm}\text {that is}\hspace {0.4cm} F_X\left (x_P\right )=P .\] If \(F_X\) is continuous and strictly increasing this equation has exactly one solution, and \[x_P=F_X^{-1}(P)=Q_X(P).\] The function \(Q_X=F_X^{-1}\), defined for \(0<P<1\), is the quantile function of \(X\).
The \(P^{th}\) quantile and the \(100P^{th}\) percentile are the same number on two scales: the \(0.25\) quantile is the \(25^{th}\) percentile. Quantile is the general term, and the quartiles, deciles and percentiles are particular quantiles, not separate objects.
Note 2.22. The requirement that \(F_X\) be strictly increasing is doing real work. If \(F_X\) is flat over an interval then \(F_X(x)=P\) holds for every \(x\) in it and the quantile is not unique; if \(F_X\) jumps past the level \(P\), as every discrete cdf does, then no \(x\) satisfies the equation at all. The definition that covers every case is \[Q_X(P)=\inf \left \{x:F_X(x)\geq P\right \},\] the smallest value at which the cdf has reached \(P\). It agrees with \(F_X^{-1}(P)\) whenever the inverse exists, and it is what any software returns.
Example 2.23. Let \(X\) be exponential with rate \(\lambda \), so \(F_X(x)=1-e^{-\lambda x}\) for \(x\geq 0\). Solving \(1-e^{-\lambda x_P}=P\) gives the quantile function \[Q_X(P)=-\dfrac {\ln (1-P)}{\lambda },\hspace {0.8cm}0<P<1 .\] The median is \(Q_X(0.5)=\ln 2/\lambda \approx 0.693/\lambda \), the lower quartile is \(Q_X(0.25)=\ln (4/3)/\lambda \approx 0.288/\lambda \) and the upper quartile is \(Q_X(0.75)=\ln 4/\lambda \approx 1.386/\lambda \). Note that the median is well below the mean \(1/\lambda \), as it must be for a distribution skewed to the right.
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