Matlab 验证部分样本空间可以预测其总体样本的性质

主验证程序的总体思想是：

1 随机产生大量的服从指数分布的随机数；

2 随机挑选出部分点的值，分别计算它们的理论平均值与真实平均值；

3 得到实验结果，画图，比较分析。

计算大量数据的平均值时，可以用期望等于其样本空间上对(1-CDF)的积分计算，shown as follows:

首先是验证程序：

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clear;

% --- Control message
CNT_Selected_Pnts = 20;

% ------------------- Not manually controlled.
meanVal = 100;
CNT_numbers = 10000;

% 1 Create serval rdm numbers following a certain model.
rdmNumbers = exprnd(meanVal, CNT_numbers, 1);

% 2 Calculate the CDF of the unmbers
min_x = min(rdmNumbers)
max_x = max(rdmNumbers)
[x,cdf_x] = funcCDF_rdm(CNT_Selected_Pnts, min_x, max_x, rdmNumbers);
plot(x, cdf_x, 'ob')

%plot(cdfNs)

% -Calculate the avgVal_ana
JF_Func = 1 - cdf_x;
AvgVal_ana = trapz(x, JF_Func)

% -Calculate the avgVal_true
AvgVal_true_lib = mean(x)
其中用到一个辅助函数 -- FuncCDF_rdm(...)，其作用是随机挑选出若干个Points，并计算出在这些点处的 CDF 值，最后返回。

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% ----
% para@1: CNT_pnts, the number of points to denote the CDF;
% para@2: Range_low, the lower bound of variable;
% para@3: Range_up, the upper bound of variable;
% para@4: arr_Vals, array of the values to be dealed with.
% ----
% Return@ x : the array of selected Pnts-value.
% Return@ CDF_Vals : the array of selected Pnts' CDF_values.
% ----
function [x, CDF_Vals] = funcCDF_rdm(CNT_pnts, Range_low, Range_up, arr_Vals)

% -- 1 Prepare the original data
data = sort( arr_Vals' ); % T', turn horizon arrays of T.
sumPnts = length(data);
Counter_every_pnt = zeros(1,CNT_pnts);

% -- 2 Count appearance times of every point
% -- 2.1_Select the random points from the original data.
idxs_rdm = randperm(sumPnts);
idx_selected = idxs_rdm(1:CNT_pnts); % !! Take the former CNT_pnts random numbers.
idx_selected = sort( idx_selected ); % !! Must be sorted.
Selected_data = data(idx_selected);
% -- 2.2_Calculate the counter between every two selected adjacent points.
for i = 1:1:sumPnts
    for selected_id = 1:1:CNT_pnts
        Current_selected_pntVal = Selected_data(selected_id);
        if ( data(1,i) <= (Range_low + Current_selected_pntVal) )
            Counter_every_pnt(1,selected_id) = Counter_every_pnt(1,selected_id) + 1; %--Inc 1 of the counter.
        end
    end
end

% -- 4 calculate the CDF_val of every point.
CDF = Counter_every_pnt(1,:)./sumPnts;

% -- 5 return the Ret-values
x = Selected_data;
CDF_Vals = CDF';