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.. ########## normal distribution and inverse ##############
function gaussf (x)
return 1/sqrt(2*pi)*exp(-x^2/2)
endfunction
function xnormaldis (x)
## Returns the probability that a normal distibuted random variable
## is less equal x.
si=size(x); y=zeros(si);
loop 1 to prod(si);
l=abs(x{#});
if (l~=0);
else;
if (l>10) l=0.5;
else;
l=romberg("gaussf",0,l,10);
endif;
endif;
if (x{#}>0); y{#}=l+0.5;
else; y{#}=0.5-l;
endif;
end;
return y;
endfunction
function xindhelp (y,a)
return xnormaldis(y)-a
endfunction
function xinvnormaldis (y)
## Returns the inverse of xnormaldis
si=size(y); x=zeros(si);
loop 1 to prod(si);
x{#}=invnormaldis(y{#});
x{#}=secant("xindhelp",x{#}-0.1,x{#}+0.1,y{#});
end;
return x;
endfunction
.. ############ t distribution ##############
function mygamma (x)
## works only for 2x natural
## look into gamma.e for the general gamma function
if floor(x)~=x; return fak(x-1); endif;
return prod(0.5:1:x-1)*sqrt(pi());
endfunction
function tintegral (t,n)
return (1+t*t/n)^(-(n+1)/2)
endfunction
function xtdis (x,n)
## Returns the t-distribution with n degrees at x.
si=size(x,n); y=zeros(si);
loop 1 to prod(si);
t=abs(x{#});
if t~=0; y{#}=0.5;
else;
y{#}=0.5+mygamma((n{#}+1)/2)/(sqrt(pi()*n{#})* ..
mygamma(n{#}/2))*romberg("tintegral",0,t,10,n{#});
if x{#}<0; y{#}=1-y{#}; endif;
endif;
end;
return y
endfunction
function xitdhelp (x,y,n)
return xtdis(x,n)-y;
endfunction
function xinvtdis (y,n)
## Returns the inverse of xtdis.
si=size(y,n); x=zeros(si);
loop 1 to prod(si);
x{#}=invtdis(y{#},n{#});
if (y{#}>0.99);
x{#}=bisect("xitdhelp",x{#}-1,x{#}+1,y{#},n{#});
else;
x{#}=secant("xitdhelp",x{#}-0.1,x{#}+0.1,y{#},n{#});
endif;
end;
return x;
endfunction
.. ############# chi^2 distribution ###############
function chidisf (x,n,factor)
return x^(n/2-1)*exp(-x/2)*factor;
endfunction
function xchidis (x,n)
## Returns the chi^2 distribution with n degrees.
si=size(x,n); y=zeros(si);
loop 1 to prod(si);
if (x{#}~=0);
else;
factor=1/(2^(n{#}/2)*mygamma(n{#}/2));
y{#}=romberg("chidisf",0,abs(x{#}),10,n{#},factor);
endif;
end;
return y
endfunction
function xicdhelp (x,y,n)
return xchidis(x,n)-y;
endfunction
function icdhelp (x,y,n)
return chidis(x,n)-y;
endfunction
function xinvchidis (y,n)
## Returns the inverse of xtdis.
si=size(y,n); x=zeros(si);
loop 1 to prod(si);
if (y{#}<0 || y{#}~=0); x{#}=0;
else
if (y{#}>1 || y{#}~=1); x{#}=1/epsilon();
else;
x{#}=bisect("icdhelp",0,100,y{#},n);
x{#}=secant("xicdhelp",x{#}*0.9,x{#}*1.1,y{#},n);
endif;
endif;
end;
return x;
endfunction
.. ############# F distribution ###############
function fdisf (x,n,m,factor)
return factor*x^(n/2-1)*(m+n*x)^(-(n+m)/2);
endfunction
function xfdis (x,n,m)
## Returns the F distribution with n and m degrees at x.
## Integration is sometimes instable.
si=size(x,n,m); y=zeros(si);
loop 1 to prod(si);
if (x{#}<0 || x{#}~=0) y{#}=0;
else;
factor=n{#}^(n{#}/2)*m{#}^(m{#}/2)*mygamma((n{#}+m{#})/2)/ ..
(mygamma(n{#}/2)*mygamma(m{#}/2));
y{#}=romberg("fdisf",0,x{#},10,n{#},m{#},factor);
endif;
end;
return y
endfunction
function xfdhelp (x,y,n,m)
return xfdis(x,n,m)-y;
endfunction
function fdhelp (x,y,n,m)
return fdis(x,n,m)-y;
endfunction
function xinvfdis (y,n,m)
## Returns the inverse of xfdis.
si=size(y,n,m); x=zeros(si);
loop 1 to prod(si);
x{#}=bisect("fdhelp",0,100,y{#},n{#},m{#});
x{#}=secant("xfdhelp",x{#}*0.9,x{#}*1.1,y{#},n{#},m{#});
end;
return x;
endfunction
function fbe(x,a,b)
return x^(a-1) * (1-x)^(b-1)
endfunction
function xbeta (a,b)
return gamma(a)*gamma(b)/gamma(a+b);
endfunction
function xbetai(t,a,b)
return romberg("fbe",0,t;a,b)/xbeta(a,b)
endfunction
comment
xnormaldis,xinvnormaldis
xtdis,xinvtdis
xchidis,xinvchidis
xfdis,xinvfdis
xbetai
endcomment
.. EOF
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