Consider the following Matlab code in which I generate some data and then extract some rows of them at random. I would like your help to understand "how" random are these draws from a statistical point of view, in the terms I explain below.
I first set some parameters
%%%%%%%%Parameters
clear
rng default
Xsup=-1:6;
Zsup=1:10;
n_m=200;
n_w=200;
R=n_m;
T=100000; %number of draws
Then I generate the data
%%%%%%%%Creation of data [XZ,etapair,zetapair,etasingle,zetasingle]
%Vector X of dimension n_mx1
idX=randi(size(Xsup,2),n_m,1); %n_mx1
X=Xsup(idX).'; %n_mx1
%Vector Z of dimension n_wx1
idZ=randi(size(Zsup,2),n_w,1);
Z=Zsup(idZ).'; %n_wx1
%Combine X and Z in a matrix XZ of dimension (n_m*n_w)x2
which lists all possible combinations of values in X and Z
[cX, cZ] = ndgrid(X,Z);
XZ = [cX(:), cZ(:)]; %(n_m*n_w)x2
%Vector etapair of dimension (n_m*n_w)x1
etapair=randn(n_m*n_w,1); %(n_m*n_w)x1
%Vector zetapair of dimension (n_m*n_w)x1
zetapair=randn(n_m*n_w,1); %(n_m*n_w)x1
%Vector etasingle of dimension n_mx1
etasingle=max(randn(n_m,R),[],2); %n_mx1
%Vector zetasingle of dimension n_wx1
zetasingle=max(randn(n_w,R),[],2); %n_wx1
Then I extract some rows at random
%%%%%%%%%%Random draws from [XZ,etapair,zetapair,etasingle,zetasingle]
idm=unidrnd(n_m,T,1); %Tx1
idw=unidrnd(n_w,T,1); %Tx1
idmupair=idm+n_m*(idw-1); %Tx1
Xsample=X(idm); %Tx1
Zsample=Z(idw); %Tx1
etapairsample=etapair(idmupair);%Tx1
zetapairsample=zetapair(idmupair);%Tx1
etasinglesample=etasingle(idm); %Tx1
zetasinglesample=zetasingle(idw); %Tx1
Let me now translate these draws into statistical terms:
For t=1,...,T
, Xsample(t)
can be thought as a realisation of a random variable X_t
For t=1,...,T
, Zsample(t)
can be thought as a realisation of a random variable Z_t
For t=1,...,T
, etapairsample(t)
can be thought as a realisation of a random variable E_t
For t=1,...,T
, zetapairsample(t)
can be thought as a realisation of a random variable Q_t
For t=1,...,T
, etasinglesample(t)
can be thought as a realisation of a random variable Y_t
For t=1,...,T
, zetasinglesample(t)
can be thought as a realisation of a random variable S_t
Since I draw realisations of these random variables at random with replacement (through the function unidrnd
) I thought I could claim that (X_1,...,X_T, Z_1,...,Z_T, E_1,...,E_T, Q_1,...,Q_T, Y_1,...,Y_T,S_1,...,S_T)
are mutually independent
As a check of this hypothetical claim, I define W_t:=-E_t-Q_t+Y_t+S_t
and empirically compute Pr(W_t<=1|X_t=5, Z_t=1)
If mutual independence holds, then Pr(W_t<=1|X_t=5, Z_t=1)=Pr(W_t<=1)
and their empirical counterparts below, named option1
and option2
, should be ALMOST the same.
%option 1: conditional probability
num1=zeros(T,1);
for h=1:T
if -etapairsample(h)-zetapairsample(h)+etasinglesample(h)+zetasinglesample(h)<=1 && Xsample(h)==5 && Zsample(h)==1
num1(h)=1;
end
end
den1=zeros(T,1);
for h=1:T
if Xsample(h)==5 && Zsample(h)==1
den1(h)=1;
end
end
option1=sum(num1)/sum(den1);
%option 2: unconditional probability
num2=zeros(T,1);
for h=1:T
if -etapairsample(h)-zetapairsample(h)+etasinglesample(h)+zetasinglesample(h)<=1
num2(h)=1;
end
end
option2=sum(num2)/T;
Question: the difference between option1
(=0.0034) and option2
(=0.0013) is referred to the "ALMOST" or I am doing something wrong?