How do I declare correlation between a random variable with beta distribution and a random variable with normal distribution?

I have a random variable with beta distribution declared:

aux_alpha = 3;

aux_beta = 813.21;

LB = 41.368542; %lower bound

D = 1337.582858; %difference between upper bound and lower bound

alpha = aux_alpha+1;

beta = aux_beta+1;

aux_fy = betarnd(alpha,beta);

fy = aux_fy*D+LB;

And a random variable with normal distribution:

Ast_mean = Ast0*b*d;

Ast_std = 0.02*Ast_mean;

Ast = normrnd(Ast_mean,Ast_std);

There is a correlation of 0.5 between them. I'm trying to test it with the code:

nsam = 10000;

rhoij = 0.5;

y1 = normrnd(0,1,nsam,1);

y2 = normrnd(0,1,nsam,1);

RY = [1 rhoij;rhoij 1];

Jzy = chol(RY,'lower');

z1 = 0;

z2 = 0;

u1 = 0;

u2 = 0;

for sam = 1:nsam

zaux = Jzy*[y1(sam);y2(sam)];

z1(sam) = zaux(1);

z2(sam) = zaux(2);

u1(sam) = normcdf(z1(sam));

u2(sam) = normcdf(z2(sam));

x1_aux(sam) = betainv(u1(sam),alpha,beta);

x1(sam) = x1_aux(sam)*D+LB;

x2(sam) = norminv(u2(sam),Ast_mean,Ast_std);

end

rho = corrcoef(x1,x2)

But rho is not returning the correct correlation (0.5). Does anyone know how to solve this? I'm in doubt especially in the part where I use betainv().

Answer 1

[Jeff Miller]

Something like this would work, though you would have to adjust rho by trial and error to get the final corr(x1,x2) value that you want:

% Just example values for some parameters--I was not sure what values you
% want.
alpha = 1.1;
beta = 4.2;
Ast_mean = 23.45;
Ast_std = 6.789;

LB = 41.368542;                      %lower bound
D = 1337.582858;                    %difference between upper bound and lower bound

nsam = 100000;  % generate a large sample so we can estimate the true correlation accurately
rho = 0.5;  % adjust this up or down to get the desired corr(x1,x2)=0.5 (or close enough)

% You shouldn't have to change anything from here
x = mvnrnd([0 0],[1 rho; rho 1],nsam);
xcdf = normcdf(x);
x1_aux = betainv(xcdf(:,1),alpha,beta);
x1 = x1_aux*D+LB;
x2 = x(:,2)*Ast_std + Ast_mean;
corr(x1,x2)

[Ana Carolina da Silva Pacheco]

Thank you!!! It worked.