# t-distribution

This section contains functions for working with noncentral Student's t-distribution. They allow to calculate density, probability, quantiles and to generate pseudo-random numbers distributed according to the law of noncentral t-distribution. The noncentral t-distribution is defined by the following formula:

where:

• x – value of the random variable
• ν –  parameter of distribution (number of degrees of freedom)
• σ – noncentrality parameter

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 MathProbabilityDensityNoncentralT Calculates the probability density function of the noncentral t-distribution MathCumulativeDistributionNoncentralT Calculates the value of the noncentral t-distribution function MathQuantileNoncentralT Calculates the value of the inverse noncentral t-distribution function for the specified probability MathRandomNoncentralT Generates a pseudorandom variable/array of pseudorandom variables distributed according to the noncentral Student's t-distribution law MathMomentsNoncentralT Calculates the theoretical numerical values of the first 4 moments of the noncentral Student's t-distribution

Example:

 #include  #include  #include  #property script_show_inputs //--- input parameters input double nu_par=30;       // the number of degrees of freedom input double delta_par=5;    // noncentrality parameter //+------------------------------------------------------------------+ //| 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 noncentral Student's t-distribution    MathRandomNoncentralT(nu_par,delta_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);    MathProbabilityDensityNoncentralT(x2,nu_par,delta_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=cells) ind=cells-1;       frequency[ind]++;      }    return (true);   } //+------------------------------------------------------------------+ //|  Calculates values for sequence generation                       | //+------------------------------------------------------------------+ void GetMaxMinStepValues(double &maxv,double &minv,double &stepv)   { //--- calculate the absolute range of the sequence to obtain the precision of normalization    double range=MathAbs(maxv-minv);    int degree=(int)MathRound(MathLog10(range)); //--- normalize the maximum and minimum values to the specified precision    maxv=NormalizeDouble(maxv,degree);    minv=NormalizeDouble(minv,degree); //--- sequence generation step is also set based on the specified precision    stepv=NormalizeDouble(MathPow(10,-degree),degree);    if((maxv-minv)/stepv<10)       stepv/=10.;   }