10.3: The Quantile Function (2024)

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

\(\{X_i: 1 \le i \le n\}\) of random variables, an independent class \(\{Y_i: 1 \le i \le n\}\) may be constructed, with each \(X_i\) and associated \(Y_i\) having the same (marginal) distribution. Quantile functions for simple random variables may be used to obtain an important Poisson approximation theorem (which we do not develop in this work). The quantile function is used to 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 (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 \(P(X \le t) = u\).

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

\(Q(u) = \text{inf } \{t: F(t) \ge u\}\) \(\forall u \in (0, 1)\)

We note

Hence, we have the important property:

(Q1) \(Q(u) \le t\) iff \(u \le F(t)\) \(\forall u \in (0, 1)\)

The property (Q1) implies the following important property:

(Q2)If \(U\)~ uniform (0, 1), then \(X = Q(U)\) has distribution function \(F_X = F\). To see this, note that \(F_X(t) = P(Q(U) \le t] = P[U \le F(t)] = F(t)\).

Property (Q2) implies that if \(F\) is any distribution function, with quantile function \(Q\), then the random variable \(X = Q(U)\), with \(U\) uniformly distributed on (0, 1), has distribution function \(F\).

Several other important properties of the quantile function may be established.

Figure 10.3.9. Graph of quantile function from graph of distribution function,

\(Q\) is left-continuous, whereas \(F\) is right-continuous.

If jumps are represented by vertical line segments, construction of the graph of \(u = Q(t)\) may be obtained by the following two step procedure:

This is illustrated in Figure 10.3.9. If jumps are represented by vertical line segments, then jumps go into flat segments and flat segments go into vertical segments.

If \(X\) is discrete with probability \(p_i\) at \(t_i\), \(1 \le i \le n\), then \(F\) has jumps in the amount \(p_i\) at each \(t_i\) and is constant between. The quantile function is a left-continuous step function having value \(t_i\) on the interval \((b_{i - 1}, b_i]\), where \(b_0 = 0\) and \(b_i = \sum_{j = 1}^{i} p_j\). This may be stated

If \(F(t_i) = b_i\), then \(Q(u) = t_i\) for \(F(t_{i - 1}) < u \le F(t_i)\)

We use the analytic characterization above in developing a number of m-functions and m-procedures.

m-procedures for a simple random variable

The basis for quantile function calculations for a simple random variable is the formula above. This is implemented in the m-function dquant, which is used as an element of several simulation procedures. To plot the quantile function, we use dquanplot which employs the stairs function and plots \(X\) vs the distribution function \(FX\). The procedure dsample employs dquant to obtain a “sample” from a population with simple distribution and to calculate relative frequencies of the various values.

X = [-2.3 -1.1 3.3 5.4 7.1 9.8];PX = 0.01*[18 15 23 19 13 12];dquanplotEnter VALUES for X XEnter PROBABILITIES for X PX % See Figure 10.3.11 for plot of resultsrand('seed',0) % Reset random number generator for referencedsampleEnter row matrix of values XEnter row matrix of probabilities PXSample size n 10000

 Value Prob Rel freq -2.3000 0.1800 0.1805 -1.1000 0.1500 0.1466 3.3000 0.2300 0.2320 5.4000 0.1900 0.1875 7.1000 0.1300 0.1333 9.8000 0.1200 0.1201Sample average ex = 3.325Population mean E[X] = 3.305Sample variance = 16.32Population variance Var[X] = 16.33

Sometimes it is desirable to know how many trials are required to reach a certain value, or one of a set of values. A pair of m-procedures are available for simulation of that problem. The first is called targetset. It calls for the population distribution and then for the designation of a “target set” of possible values. The second procedure, targetrun, calls for the number of repetitions of the experiment, and asks for the number of members of the target set to be reached. After the runs are made, various statistics on the runs are calculated and displayed.

X = [-1.3 0.2 3.7 5.5 7.3]; % Population valuesPX = [0.2 0.1 0.3 0.3 0.1]; % Population probabilitiesE = [-1.3 3.7]; % Set of target statestargetsetEnter population VALUES XEnter population PROBABILITIES PXThe set of population values is -1.3000 0.2000 3.7000 5.5000 7.3000Enter the set of target values ECall for targetrun

rand('seed',0) % Seed set for possible comparisontargetrunEnter the number of repetitions 1000The target set is -1.3000 3.7000Enter the number of target values to visit 2The average completion time is 6.32The standard deviation is 4.089The minimum completion time is 2The maximum completion time is 30To view a detailed count, call for D.The first column shows the various completion times;the second column shows the numbers of trials yielding those times% Figure 10.6.4 shows the fraction of runs requiring t steps or less

Figure 10.3.12. Fraction of runs requiring \(t\) steps or less.

m-procedures for distribution functions

A procedure dfsetup utilizes the distribution function to set up an approximate simple distribution. The m-procedure quanplot is used to plot the quantile function. This procedure is essentially the same as dquanplot, except the ordinary plot function is used in the continuous case whereas the plotting function stairs is used in the discrete case. The m-procedure qsample is used to obtain a sample from the population. Since there are so many possible values, these are not displayed as in the discrete case.

F = '0.4*(t + 1).*(t < 0) + (0.6 + 0.4*t).*(t >= 0)'; % StringdfsetupDistribution function F is entered as a stringvariable, either defined previously or upon callEnter matrix [a b] of X-range endpoints [-1 1]Enter number of X approximation points 1000Enter distribution function F as function of t FDistribution is in row matrices X and PXquanplotEnter row matrix of values XEnter row matrix of probabilities PXProbability increment h 0.01 % See Figure 10.3.13 for plotqsampleEnter row matrix of X values XEnter row matrix of X probabilities PXSample size n 1000Sample average ex = -0.004146Approximate population mean E(X) = -0.0004002 % Theoretical = 0Sample variance vx = 0.25Approximate population variance V(X) = 0.2664

m-procedures for density functions

An m- procedure acsetup is used to obtain the simple approximate distribution. This is essentially the same as the procedure tuappr, except that the density function is entered as a string variable. Then the procedures quanplot and qsample are used as in the case of distribution functions.

acsetupDensity f is entered as a string variable.either defined previously or upon call.Enter matrix [a b] of x-range endpoints [0 3]Enter number of x approximation points 1000Enter density as a function of t '(t.^2).*(t<1) + (1- t/3).*(1<=t)'Distribution is in row matrices X and PXquanplotEnter row matrix of values XEnter row matrix of probabilities PXProbability increment h 0.01 % See Figure 10.3.14 for plotrand('seed',0)qsampleEnter row matrix of values XEnter row matrix of probabilities PXSample size n 1000Sample average ex = 1.352Approximate population mean E(X) = 1.361 % Theoretical = 49/36 = 1.3622Sample variance vx = 0.3242Approximate population variance V(X) = 0.3474 % Theoretical = 0.3474