//+------------------------------------------------------------------+
//|                                    NormalDistributionExample.mq5 |
//|                        Copyright 2016, MetaQuotes Software Corp. |
//|                                             https://www.mql5.com |
//+------------------------------------------------------------------+
#property copyright "Copyright 2000-2024, MetaQuotes Ltd."
#property link      "https://www.mql5.com"
#property version   "1.00"
//--- include the functions for calculating the normal distribution
#include <Math\Stat\Normal.mqh>
//+------------------------------------------------------------------+
//| Script program start function                                    |
//+------------------------------------------------------------------+
void OnStart()
  {
//--- set the parameters of the normal distribution
   double mu=5.0;
   double sigma=1.0;
   PrintFormat("Normal distribution with parameters mu=%G and sigma=%G, calculation examples:",mu,sigma);
//--- set the interval
   double x1=mu-sigma;
   double x2=mu+sigma;
//--- variables for probability calculation
   double cdf1,cdf2,probability;
//--- variables for error codes
   int error_code1,error_code2;
//--- calculate the values of distribution functions
   cdf1=MathCumulativeDistributionNormal(x1,mu,sigma,error_code1);
   cdf2=MathCumulativeDistributionNormal(x2,mu,sigma,error_code2);
//--- check the error codes
   if(error_code1==ERR_OK && error_code2==ERR_OK)
     {
      //--- calculate probability of a random variable in the range
      probability=cdf2-cdf1;
      //--- output the result
      PrintFormat("1. Calculate probability of a random variable within the range of %.5f<x<%.5f",x1,x2);
      PrintFormat("  Answer: Probability = %5.8f",probability);
     }

//--- Find the value range of random variable x, corresponding to the 95% confidence level
   probability=0.95; //  set the confidence probability
//--- set the probabilities at the interval bounds
   double p1=(1.0-probability)*0.5;
   double p2=probability+(1.0-probability)*0.5;
//--- calculate the interval bounds
   x1=MathQuantileNormal(p1,mu,sigma,error_code1);
   x2=MathQuantileNormal(p2,mu,sigma,error_code2);
//--- check the error codes
   if(error_code1==ERR_OK && error_code2==ERR_OK)
     {
      //--- output the result
      PrintFormat("2. For confidence interval = %.2f, find the range of random variable",probability);
      PrintFormat("  Answer: range is  %5.8f <= x <=%5.8f",x1,x2);
     }

   PrintFormat("3. Compute the first 4 calculated and theoretical moments of the distribution");
//--- Generate an array of random numbers, calculate the first 4 moments and compare with the theoretical values
   int data_count=1000000;  // set the number of values and prepare an array
   double data[];
   ArrayResize(data,data_count);
//--- generate random values and store them into the array
   for(int i=0; i<data_count; i++)
     {
      data[i]=MathRandomNormal(mu,sigma,error_code1);
     }
//--- set the index of the initial value and the amount of data for calculation
   int start=0;
   int count=data_count;
//--- calculate the first 4 moments of the generated values
   double mean=MathMean(data,start,count);
   double variance=MathVariance(data,start,count);
   double skewness=MathSkewness(data,start,count);
   double kurtosis=MathKurtosis(data,start,count);
//--- variables for the theoretical moments
   double normal_mean=0;
   double normal_variance=0;
   double normal_skewness=0;
   double normal_kurtosis=0;
//--- display the values of the calculated moments
   PrintFormat("            Mean           Variance          Skewness           Kurtosis");
   PrintFormat("Calculated  %.10f   %.10f      %.10f      %.10f",mean,variance,skewness,kurtosis);
//--- calculate the theoretical values of the moments and compare them with the obtained values
   if(MathMomentsNormal(mu,sigma,normal_mean,normal_variance,normal_skewness,normal_kurtosis,error_code1))
     {
      PrintFormat("Theoretical %.10f   %.10f      %.10f       %.10f",normal_mean,normal_variance,normal_skewness,normal_kurtosis);
      PrintFormat("Difference  %.10f   %.10f      %.10f       %.10f",mean-normal_mean,variance-normal_variance,skewness-normal_skewness,kurtosis-normal_kurtosis);
     }
  }