# 10.2 The quantile function

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If F is a probability distribution function, the associated quantile function Q is essentially an inverse of F. The quantile function is defined on the unit interval (0,1). For F continuous and strictly increasing at t, then Q(u)=t iff F(t)=u. Thus, if u is a probability value, t=Q(u) is the value of t for which P(X≤t)=u. As an application, one may generate independent classes with prescribed distributions. Matlab procedures are developed for a variety of problems involving quantile functions.

## The quantile function

The quantile function for a probability distribution has many uses in both the theory and application of probability. If F is a probability distribution function, the quantile function may be used to “construct” a random variable having F as its distributions function. This fact serves as the basis of a method of simulating the“sampling” from an arbitrary distribution with the aid of a random number generator . Also, given any finite class

$\left\{{X}_{i}:1\le i\le n\right\}$ of random variables, an independent class $\left\{{Y}_{i}:1\le i\le n\right\}$ may be constructed, with each X i and associated Y i having the same (marginal) distribution. Quantile functions for simple random variables maybe used to obtain an important Poisson approximation theorem (which we do not develop in this work). The quantile function is usedto derive a number of useful special forms for mathematical expectation.

General concept—properties, and examples

If F is a probability distribution function, the associated quantile function Q is essentially an inverse of F . The quantile function is defined on the unit interval $\left(0,\phantom{\rule{0.166667em}{0ex}}1\right)$ . For F continuous and strictly increasing at t , then $Q\left(u\right)=t$ iff $F\left(t\right)=u$ . Thus, if u is a probability value, $t=Q\left(u\right)$ is the value of t for which $P\left(X\le t\right)=u$ .

## The weibull distribution $\left(3,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}0\right)$

$u=F\left(t\right)=1-{e}^{-3{t}^{2}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t\ge 0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}⇒\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t=Q\left(u\right)=\sqrt{-ln\left(1-u\right)/3}$

## The normal distribution

The m-function norminv , based on the MATLAB function erfinv (inverse error function), calculates values of Q for the normal distribution.

The restriction to the continuous case is not essential. We consider a general definition which applies to any probability distribution function.

Definition : If F is a function having the properties of a probability distribution function, then the quantile function for F is given by

$Q\left(u\right)=inf\left\{t:\phantom{\rule{0.166667em}{0ex}}F\left(t\right)\ge u\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}u\in \left(0,\phantom{\rule{0.166667em}{0ex}}1\right)$

We note

• If $F\left({t}^{*}\right)\ge u*$ , then ${t}^{*}\ge inf\left\{t:\phantom{\rule{0.166667em}{0ex}}F\left(t\right)\ge {u}^{*}\right\}=Q\left({u}^{*}\right)$
• If $F\left({t}^{*}\right) , then ${t}^{*}

Hence, we have the important property:

(Q1) $Q\left(u\right)\le t$ iff $u\le F\left(t\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}u\in \left(0,\phantom{\rule{0.166667em}{0ex}}1\right)$ .

The property (Q1) implies the following important property:

(Q2) If $U\sim$ uniform $\left(0,\phantom{\rule{0.166667em}{0ex}}1\right)$ , then $X=Q\left(U\right)$ has distribution function ${F}_{X}=F$ . To see this, note that ${F}_{X}\left(t\right)=P\left[Q\left(U\right)\le t\right]=P\left[U\le F\left(t\right)\right]=F\left(t\right)$ .

Property (Q2) implies that if F is any distribution function, with quantile function Q , then the random variable $X=Q\left(U\right)$ , with U uniformly distributed on $\left(0,\phantom{\rule{0.166667em}{0ex}}1\right)$ , has distribution function F .

## Independent classes with prescribed distributions

Suppose $\left\{{X}_{i}:1\le i\le n\right\}$ is an arbitrary class of random variables with corresponding distribution functions $\left\{{F}_{i}:1\le i\le n\right\}$ . Let $\left\{{Q}_{i}:1\le i\le n\right\}$ be the respective quantile functions. There is always an independent class $\left\{{U}_{i}:1\le i\le n\right\}$ iid uniform $\left(0,\phantom{\rule{0.166667em}{0ex}}1\right)$ (marginals for the joint uniform distribution on the unit hypercube with sides $\left(0,\phantom{\rule{0.166667em}{0ex}}1\right)$ ). Then the random variables ${Y}_{i}={Q}_{i}\left({U}_{i}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}1\le i\le n$ , form an independent class with the same marginals as the X i .

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