# Cauchy distribution

This section contains functions for working with Cauchy distribution. They allow to calculate density, probability, quantiles and to generate pseudo-random numbers distributed according to the Cauchy law. The Cauchy distribution is defined by the following formula:

where:

• x — value of the random variable
• a — mean parameter of the distribution
• b — scale parameter of the distribution

In addition to the calculation of the individual random variables, the library also implements the ability to work with arrays of random variables.

 Function Description MathProbabilityDensityCauchy Calculates the probability density function of the Cauchy distribution MathCumulativeDistributionCauchy Calculates the value of the Cauchy probability distribution function MathQuantileCauchy Calculates the value of the inverse Cauchy distribution function for the specified probability MathRandomCauchy Generates a pseudorandom variable/array of pseudorandom variables distributed according to the Cauchy law MathMomentsCauchy Calculates the theoretical numerical values of the first 4 moments of the Cauchy distribution

Example:

 #include  #include  #include  #property script_show_inputs //--- input parameters input double a_par=-2;      // mean parameter of the distribution input double b_par=1;       // scale parameter of the distribution //+------------------------------------------------------------------+ //| Script program start function                                    | //+------------------------------------------------------------------+ void OnStart()   { //--- hide the price chart    ChartSetInteger(0,CHART_SHOW,false); //--- initialize the random number generator      MathSrand(GetTickCount()); //--- generate a sample of the random variable    long chart=0;    string name="GraphicNormal";    int n=1000000;       // the number of values in the sample    int ncells=51;       // the number of intervals in the histogram    double x[];          // centers of the histogram intervals    double y[];          // the number of values from the sample falling within the interval    double data[];       // sample of random values    double max,min;      // the maximum and minimum values in the sample //--- obtain a sample from the Cauchy distribution    MathRandomCauchy(a_par,b_par,n,data); //--- calculate the data to plot the histogram    CalculateHistogramArray(data,x,y,max,min,ncells); //--- obtain the sequence boundaries and the step for plotting the theoretical curve    double step;    GetMaxMinStepValues(max,min,step);    step=MathMin(step,(max-min)/ncells); //--- obtain the theoretically calculated data at the interval of [min,max]    double x2[];    double y2[];    MathSequence(min,max,step,x2);    MathProbabilityDensityCauchy(x2,a_par,b_par,false,y2); //--- set the scale    double theor_max=y2[ArrayMaximum(y2)];    double sample_max=y[ArrayMaximum(y)];    double k=sample_max/theor_max;    for(int i=0; i