> restart:
> with(Statistics):
> pdf:=PDF(RandomVariable(Normal(mu,sigma)),x);
> cdf:=CDF(RandomVariable(Normal(mu,sigma)),x);
> pdf_cond:=subs(x=mu+k*sigma+t,pdf)/subs(x=mu+k*sigma,cdf);

                                                 2
                               1/2       (x - mu)
                              2    exp(- ---------)
                                                2
                                         2 sigma
                   pdf := 1/2 ---------------------
                                     1/2
                                   Pi    sigma


                                       1/2
                                      2    (x - mu)
                 cdf := 1/2 + 1/2 erf(-------------)
                                         2 sigma


                                                     2
                              1/2       (k sigma + t)
                             2    exp(- --------------)
                                                  2
                                           2 sigma
         pdf_cond := 1/2 -----------------------------------
                                     /               1/2   \
                           1/2       |              2    k |
                         Pi    sigma |1/2 + 1/2 erf(------)|
                                     \                2    /

> series(pdf_cond,t=infinity);

                                  2             2
                      1/2        k             t
                     2    exp(- ----) exp(- --------)
                                 2                 2
                                            2 sigma
          1/2 ----------------------------------------------
                          /               1/2   \
                1/2       |              2    k |      k t
              Pi    sigma |1/2 + 1/2 erf(------)| exp(-----)
                          \                2    /     sigma

> with(LinearAlgebra):
> INF:=10.0:
> x1:=-1.0:
> dx1:=0.01:
> for dim from 2 to 15 do
>   M:=Matrix(dim):
>   for i from 1 to dim do
>     M[i,i]:=1;
>   od:
>   for i from 1 to dim-1 do
>     M[i,i+1]:=1/2;
>     M[i+1,i]:=1/2;
>   od:
>   V:=Vector([seq(x[i],i=1..dim)]):
>  
> pdf:=1/(2*Pi)^(dim/2)/sqrt(Determinant(M))*exp(-1/2*Transpose(V).M^(-1
> ).V);
>   formula:=Int(pdf,x[1]=x1..x1+dx1);
>   for i from 2 to dim do
>     formula:=Int(formula,x[i]=-INF..x1);
>   od:
>   val:=evalf(formula);
>   print([dim,val]);
> od:

                         [2, 0.0006830359542]


                         [3, 0.0002451522253]


                        [4, 0.00007881966467]


                        [5, 0.00002645490603]


         -1.0    -1.0    -1.0    -1.0    -1.0    -0.99
        /       /       /       /       /       /
       |       |       |       |       |       |
  [6,  |       |       |       |       |       |       0.01218993421
       |       |       |       |       |       |
      /       /       /       /       /       /
        -10.0   -10.0   -10.0   -10.0   -10.0   -1.0

        exp(-1. x[1] (0.8571428571 x[1] - 0.7142857143 x[2]

         + 0.5714285714 x[3] - 0.4285714286 x[4] + 0.2857142857 x[5]

         - 0.1428571429 x[6]) - 1. x[2] (-0.7142857143 x[1]

         + 1.428571429 x[2] - 1.142857143 x[3] + 0.8571428571 x[4]

         - 0.5714285714 x[5] + 0.2857142857 x[6]) - 1. x[3] (

        0.5714285714 x[1] - 1.142857143 x[2] + 1.714285714 x[3]

         - 1.285714286 x[4] + 0.8571428571 x[5] - 0.4285714286 x[6])

         - 1. x[4] (-0.4285714286 x[1] + 0.8571428571 x[2]

         - 1.285714286 x[3] + 1.714285714 x[4] - 1.142857143 x[5]

         + 0.5714285714 x[6]) - 1. x[5] (0.2857142857 x[1]

         - 0.5714285714 x[2] + 0.8571428571 x[3] - 1.142857143 x[4]

         + 1.428571429 x[5] - 0.7142857143 x[6]) - 1. x[6] (

        -0.1428571429 x[1] + 0.2857142857 x[2] - 0.4285714286 x[3]

         + 0.5714285714 x[4] - 0.7142857143 x[5] + 0.8571428571 x[6]))

        dx[1] dx[2] dx[3] dx[4] dx[5] dx[6]]


         -1.0    -1.0    -1.0    -1.0    -1.0    -1.0    -0.99
        /       /       /       /       /       /       /
       |       |       |       |       |       |       |
  [7,  |       |       |       |       |       |       |
       |       |       |       |       |       |       |
      /       /       /       /       /       /       /
        -10.0   -10.0   -10.0   -10.0   -10.0   -10.0   -1.0

        0.006433250340 exp(-1. x[1] (0.8750000000 x[1]

         - 0.7500000000 x[2] + 0.6250000000 x[3] - 0.5000000000 x[4]

         + 0.3750000000 x[5] - 0.2500000000 x[6] + 0.1250000000 x[7])

         - 1. x[2] (-0.7500000000 x[1] + 1.500000000 x[2]

         - 1.250000000 x[3] + x[4] - 0.7500000000 x[5]

         + 0.5000000000 x[6] - 0.2500000000 x[7]) - 1. x[3] (

        0.6250000000 x[1] - 1.250000000 x[2] + 1.875000000 x[3]

         - 1.500000000 x[4] + 1.125000000 x[5] - 0.7500000000 x[6]

         + 0.3750000000 x[7]) - 1. x[4] (-0.5000000000 x[1] + x[2]

         - 1.500000000 x[3] + 2. x[4] - 1.500000000 x[5] + x[6]

         - 0.5000000000 x[7]) - 1. x[5] (0.3750000000 x[1]

         - 0.7500000000 x[2] + 1.125000000 x[3] - 1.500000000 x[4]

         + 1.875000000 x[5] - 1.250000000 x[6] + 0.6250000000 x[7])

         - 1. x[6] (-0.2500000000 x[1] + 0.5000000000 x[2]

         - 0.7500000000 x[3] + x[4] - 1.250000000 x[5]

         + 1.500000000 x[6] - 0.7500000000 x[7]) - 1. x[7] (

        0.1250000000 x[1] - 0.2500000000 x[2] + 0.3750000000 x[3]

         - 0.5000000000 x[4] + 0.6250000000 x[5] - 0.7500000000 x[6]

         + 0.8750000000 x[7])) dx[1] dx[2] dx[3] dx[4] dx[5] dx[6] d

        x[7]]


         -1.0    -1.0    -1.0    -1.0    -1.0    -1.0    -1.0
        /       /       /       /       /       /       /
       |       |       |       |       |       |       |
  [8,  |       |       |       |       |       |       |
       |       |       |       |       |       |       |
      /       /       /       /       /       /       /
        -10.0   -10.0   -10.0   -10.0   -10.0   -10.0   -10.0

           -0.99
          /
         |
         |       0.003421994083 exp(-1. x[1] (0.8888888889 x[1]
         |
        /
          -1.0

         - 0.7777777778 x[2] + 0.6666666667 x[3] - 0.5555555556 x[4]

         + 0.4444444444 x[5] - 0.3333333333 x[6] + 0.2222222222 x[7]

         - 0.1111111111 x[8]) - 1. x[2] (-0.7777777778 x[1]

         + 1.555555556 x[2] - 1.333333333 x[3] + 1.111111111 x[4]

         - 0.8888888889 x[5] + 0.6666666667 x[6] - 0.4444444444 x[7]

         + 0.2222222222 x[8]) - 1. x[3] (0.6666666667 x[1]

         - 1.333333333 x[2] + 2. x[3] - 1.666666667 x[4]

         + 1.333333333 x[5] - 1. x[6] + 0.6666666667 x[7]

         - 0.3333333333 x[8]) - 1. x[4] (-0.5555555556 x[1]

         + 1.111111111 x[2] - 1.666666667 x[3] + 2.222222222 x[4]

         - 1.777777778 x[5] + 1.333333333 x[6] - 0.8888888889 x[7]

         + 0.4444444444 x[8]) - 1. x[5] (0.4444444444 x[1]

         - 0.8888888889 x[2] + 1.333333333 x[3] - 1.777777778 x[4]

         + 2.222222222 x[5] - 1.666666667 x[6] + 1.111111111 x[7]

         - 0.5555555556 x[8]) - 1. x[6] (-0.3333333333 x[1]

         + 0.6666666667 x[2] - 1. x[3] + 1.333333333 x[4]

         - 1.666666667 x[5] + 2. x[6] - 1.333333333 x[7]

         + 0.6666666667 x[8]) - 1. x[7] (0.2222222222 x[1]

         - 0.4444444444 x[2] + 0.6666666667 x[3] - 0.8888888889 x[4]

         + 1.111111111 x[5] - 1.333333333 x[6] + 1.555555556 x[7]

         - 0.7777777778 x[8]) - 1. x[8] (-0.1111111111 x[1]

         + 0.2222222222 x[2] - 0.3333333333 x[3] + 0.4444444444 x[4]

         - 0.5555555556 x[5] + 0.6666666667 x[6] - 0.7777777778 x[7]

         + 0.8888888889 x[8])) dx[1] dx[2] dx[3] dx[4] dx[5] dx[6] d

        x[7] dx[8]]


         -1.0    -1.0    -1.0    -1.0    -1.0    -1.0    -1.0    -1.0
        /       /       /       /       /       /       /       /
       |       |       |       |       |       |       |       |
  [9,  |       |       |       |       |       |       |       |
       |       |       |       |       |       |       |       |
      /       /       /       /       /       /       /       /
        -10.0   -10.0   -10.0   -10.0   -10.0   -10.0   -10.0   -10.0

           -0.99
          /
         |
         |       0.001831578650 exp(-1. x[1] (0.9000000000 x[1]
         |
        /
          -1.0

         - 0.8000000000 x[2] + 0.7000000000 x[3] - 0.6000000000 x[4]

         + 0.5000000000 x[5] - 0.4000000000 x[6] + 0.3000000000 x[7]

         - 0.2000000000 x[8] + 0.1000000000 x[9]) - 1. x[2] (

        -0.8000000000 x[1] + 1.600000000 x[2] - 1.400000000 x[3]

         + 1.200000000 x[4] - 1. x[5] + 0.8000000000 x[6]

         - 0.6000000000 x[7] + 0.4000000000 x[8] - 0.2000000000 x[9])

         - 1. x[3] (0.7000000000 x[1] - 1.400000000 x[2]

         + 2.100000000 x[3] - 1.800000000 x[4] + 1.500000000 x[5]

         - 1.200000000 x[6] + 0.9000000000 x[7] - 0.6000000000 x[8]

         + 0.3000000000 x[9]) - 1. x[4] (-0.6000000000 x[1]

         + 1.200000000 x[2] - 1.800000000 x[3] + 2.400000000 x[4]

         - 2. x[5] + 1.600000000 x[6] - 1.200000000 x[7]

         + 0.8000000000 x[8] - 0.4000000000 x[9]) - 1. x[5] (

        0.5000000000 x[1] - 1. x[2] + 1.500000000 x[3] - 2. x[4]

         + 2.500000000 x[5] - 2. x[6] + 1.500000000 x[7] - 1. x[8]

         + 0.5000000000 x[9]) - 1. x[6] (-0.4000000000 x[1]

         + 0.8000000000 x[2] - 1.200000000 x[3] + 1.600000000 x[4]

         - 2. x[5] + 2.400000000 x[6] - 1.800000000 x[7]

         + 1.200000000 x[8] - 0.6000000000 x[9]) - 1. x[7] (

        0.3000000000 x[1] - 0.6000000000 x[2] + 0.9000000000 x[3]

         - 1.200000000 x[4] + 1.500000000 x[5] - 1.800000000 x[6]

         + 2.100000000 x[7] - 1.400000000 x[8] + 0.7000000000 x[9])

         - 1. x[8] (-0.2000000000 x[1] + 0.4000000000 x[2]

         - 0.6000000000 x[3] + 0.8000000000 x[4] - 1. x[5]

         + 1.200000000 x[6] - 1.400000000 x[7] + 1.600000000 x[8]

         - 0.8000000000 x[9]) - 1. x[9] (0.1000000000 x[1]

         - 0.2000000000 x[2] + 0.3000000000 x[3] - 0.4000000000 x[4]

         + 0.5000000000 x[5] - 0.6000000000 x[6] + 0.7000000000 x[7]

         - 0.8000000000 x[8] + 0.9000000000 x[9])) dx[1] dx[2] dx[3]

        dx[4] dx[5] dx[6] dx[7] dx[8] dx[9]]


          -1.0    -1.0    -1.0    -1.0    -1.0    -1.0    -1.0
         /       /       /       /       /       /       /
        |       |       |       |       |       |       |
  [10,  |       |       |       |       |       |       |
        |       |       |       |       |       |       |
       /       /       /       /       /       /       /
         -10.0   -10.0   -10.0   -10.0   -10.0   -10.0   -10.0

           -1.0    -1.0    -0.99
          /       /       /
         |       |       |
         |       |       |       0.0009852678091 exp(-1. x[1] (
         |       |       |
        /       /       /
          -10.0   -10.0   -1.0

        0.9090909091 x[1] - 0.8181818182 x[2] + 0.7272727273 x[3]

         - 0.6363636364 x[4] + 0.5454545455 x[5] - 0.4545454545 x[6]

         + 0.3636363636 x[7] - 0.2727272727 x[8] + 0.1818181818 x[9]

         - 0.09090909091 x[10]) - 1. x[2] (-0.8181818182 x[1]

         + 1.636363636 x[2] - 1.454545455 x[3] + 1.272727273 x[4]

         - 1.090909091 x[5] + 0.9090909091 x[6] - 0.7272727273 x[7]

         + 0.5454545455 x[8] - 0.3636363636 x[9] + 0.1818181818 x[10]

        ) - 1. x[3] (0.7272727273 x[1] - 1.454545455 x[2]

         + 2.181818182 x[3] - 1.909090909 x[4] + 1.636363636 x[5]

         - 1.363636364 x[6] + 1.090909091 x[7] - 0.8181818182 x[8]

         + 0.5454545455 x[9] - 0.2727272727 x[10]) - 1. x[4] (

        -0.6363636364 x[1] + 1.272727273 x[2] - 1.909090909 x[3]

         + 2.545454545 x[4] - 2.181818182 x[5] + 1.818181818 x[6]

         - 1.454545455 x[7] + 1.090909091 x[8] - 0.7272727273 x[9]

         + 0.3636363636 x[10]) - 1. x[5] (0.5454545455 x[1]

         - 1.090909091 x[2] + 1.636363636 x[3] - 2.181818182 x[4]

         + 2.727272727 x[5] - 2.272727273 x[6] + 1.818181818 x[7]

         - 1.363636364 x[8] + 0.9090909091 x[9] - 0.4545454545 x[10])

         - 1. x[6] (-0.4545454545 x[1] + 0.9090909091 x[2]

         - 1.363636364 x[3] + 1.818181818 x[4] - 2.272727273 x[5]

         + 2.727272727 x[6] - 2.181818182 x[7] + 1.636363636 x[8]

         - 1.090909091 x[9] + 0.5454545455 x[10]) - 1. x[7] (

        0.3636363636 x[1] - 0.7272727273 x[2] + 1.090909091 x[3]

         - 1.454545455 x[4] + 1.818181818 x[5] - 2.181818182 x[6]

         + 2.545454545 x[7] - 1.909090909 x[8] + 1.272727273 x[9]

         - 0.6363636364 x[10]) - 1. x[8] (-0.2727272727 x[1]

         + 0.5454545455 x[2] - 0.8181818182 x[3] + 1.090909091 x[4]

         - 1.363636364 x[5] + 1.636363636 x[6] - 1.909090909 x[7]

         + 2.181818182 x[8] - 1.454545455 x[9] + 0.7272727273 x[10])

         - 1. x[9] (0.1818181818 x[1] - 0.3636363636 x[2]

         + 0.5454545455 x[3] - 0.7272727273 x[4] + 0.9090909091 x[5]

         - 1.090909091 x[6] + 1.272727273 x[7] - 1.454545455 x[8]

         + 1.636363636 x[9] - 0.8181818182 x[10]) - 1. x[10] (

        -0.09090909091 x[1] + 0.1818181818 x[2] - 0.2727272727 x[3]

         + 0.3636363636 x[4] - 0.4545454545 x[5] + 0.5454545455 x[6]

         - 0.6363636364 x[7] + 0.7272727273 x[8] - 0.8181818182 x[9]

         + 0.9090909091 x[10])) dx[1] dx[2] dx[3] dx[4] dx[5] dx[6] d

        x[7] dx[8] dx[9] dx[10]]

Warning, computation interrupted

> 
