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|
> restart:
> with(LinearAlgebra):
> dim:=3;
> nu:=2;
> Sigma:=<<1,1/2,1/4>|<1/2,1,1/2>|<1/4,1/2,1>>;
> x:=<x1,x2,x3>;
> p:=GAMMA((dim+nu)/2)/(GAMMA(nu/2)*nu^(dim/2)*Pi^(dim/2)*Determinant(Si
> gma)^(1/2)*(1+1/nu*Transpose(x).MatrixInverse(Sigma).x)^((dim+nu)/2));
dim := 3
nu := 2
[ 1 1/2 1/4]
[ ]
Sigma := [1/2 1 1/2]
[ ]
[1/4 1/2 1 ]
[x1]
[ ]
x := [x2]
[ ]
[x3]
1/2 1/2 1/2 / / / /2 x1 x2 \
p := 2 9 16 / |48 Pi |1 + x1 |---- - ----|
/ \ \ \ 3 3 /
/ x1 5 x2 x3 \ / x2 2 x3\\5/2\
+ x2 |- ---- + ---- - ----| + x3 |- ---- + ----|| |
\ 3 6 3 / \ 3 3 // /
> for n from 1 to 8 do
>
> disp([n,evalf[n](Int(Int(Int(p,x1=-infinity..1/2),x2=-infinity..1),x3=
> -infinity..3/2))]);
> od;
disp([1, 0.6])
disp([2, 0.55])
disp([3, 0.551])
disp([4, 0.5509])
disp([5, 0.55090])
disp([6, 0.550895])
disp([7, 0.5508953])
disp([8, 0.55089531])
> with(Statistics):
> X:=RandomVariable(StudentT(NU)):
> pt:=PDF(X,t);
NU
GAMMA(1/2 + ----)
2
pt := ---------------------------------------------
/ NU \
|1/2 + ----|
\ 2 /
/ 2 \
1/2 NU | t |
(Pi NU) GAMMA(----) |1 + ----|
2 \ NU /
> simplify((pt*sqrt(2+t^2)^3),symbolic);
/ NU \
/ NU \ |- 1/2 - ----|
|----| \ 2 /
NU \ 2 / 2 3/2 2 /
GAMMA(1/2 + ----) NU (2 + t ) (NU + t ) /
2 /
/ 1/2 NU \
|Pi GAMMA(----)|
\ 2 /
> convert(1/Beta(1/2,NU/2),GAMMA);
NU
GAMMA(1/2 + ----)
2
-----------------
1/2 NU
Pi GAMMA(----)
2
> coef:=1/Pi:
> for i from 1 to 101-1 do
> if (i mod 2 = 0) and (i>2) then
> coef := coef * (i-1) / (i-2);
> fi;
> od;
> evalf(coef);
2.533404109
> (coef/Beta(1/2,101/2-1));
99
---
2
Pi
>
|
