Lösungen/Solutions/F2S2Ue/Ex08

L = Lektion/ Leçon   -- Sprechstunde / heures de consultation
W = Work / Arbeit / Travail
A=Anhang/Annexe/Appendice

Ueb/Ex 1      W

a

f[x_, y_] := 
        1/(2 Pi) E^(-1/2 (x^2 + y^2));

Plot3D[f[x, y], {x, -3, 3}, {y, -3, 3}];

μX = 3;  μY = 2;    σX = 3;  σY = 
      2/3;  ϱXY = 1/2;

f1[x_, y_] := 
        1/(2 Pi σX σY Sqrt[1 - ϱXY^2]) 
          E^((-1/(2 (1 - ϱXY^2)) (((x - μX)^2)/(σX^2) + ((y - μY)^2)/(σY^2) -  2 ϱXY  (x - μX) (y - μY)/(σX σY))));

?f1

Plot3D[f1[x, y], {x, -2, 8}, {y, -2, 8}, 
        PlotRange -> {0, 0.1}];

b

Integrate[
      Integrate[f[x, y], {x, -a, a}], {y, -a, a}]

Integrate[
      Integrate[f[x, y], {x, -Infinity, Infinity}], {y, -Infinity, 
        Infinity}]

Integrate[
        Integrate[f[x, y], {x, -a, a}], {y, -a, a}] /. {a -> 
          Infinity}

Integrate[
      Integrate[f[x, y], {x, -Infinity, Infinity}], {y, -Infinity, 
        Infinity}]

Integrate[
      Integrate[f1[x, y], {x, -Infinity, Infinity}], {y, -Infinity, 
        Infinity}]

Integrate[f1[x, y], {x, -Infinity, Infinity}]

Integrate[f1[x, y], {y, -Infinity, Infinity}]

c

F[x_, y_] := 
      Integrate[
        Integrate[f[u, v], {u, -Infinity, x}], {v, -Infinity, 
          y}]

F[0, 0]

F[0, Infinity]

F[Infinity, 0]

F[Infinity, Infinity]

F1[x_, y_] := 
      Integrate[
        Integrate[f1[u, v], {u, -Infinity, x}], {v, -Infinity, 
          y}]

NF1[x_, y_] := 
      NIntegrate[
        Integrate[f1[u, v], {u, -Infinity, x}], {v, -Infinity, 
          y}]

NF1[3, 2]

Ueb/Ex 2      W

a

Xbar=X1+X2; 
μ=μ1=μ2;
σ^2=σ1^2 / 2 = σ2^2 / 2;

b

Remove[f2, F2]

f2[x_, σ_, μ_] := 
        1/(Sqrt[2 Pi] σ) 
          E^(-1/(2) ((x - μ)^2/σ^2));

F2[x_, σ_, μ_] := 
      Integrate[f2[u, σ, μ], {u, -Infinity, x}]

F2[Infinity, σ, μ] // Simplify

F2[Infinity, 4, 5]

F2[μ]

f3[x_, σ_, μ_] := 
        1/(Sqrt[2 Pi] c1 σ) 
          E^(-1/(2) ((c1 x + 
                          c2 - (c1 μ + 
                              c2))^2/(c1 σ)^2));

F3[x_, σ_, μ_] := 
      Integrate[
        f3[u, σ, μ]* 
          Evaluate[D[c1 u + c2, u]], {u, -Infinity, x}]

F3[Infinity, 4, 5]

Ueb/Ex 3      W

a

k[n_] := 1/(2^(n/2) Gamma[n/2])

Table[k[n], {n, 0, 10}]

f4[x_, n_] := k[n] x^((n - 2)/2) E^(-x/2)

Plot[{f4[x, 10], f4[x, 5]}, {x, 0, 10}];

F4[x_, n_] := Integrate[f4[u, n], {u, 0, x}]

F4[2, 5]

F4[2, 5] // N

b

Remove[f5, f6, F5, F6]

f5[z_, 
        n_] := Gamma[(n + 1)/2]/Sqrt[n Pi]/
          Gamma[n/2] /(1 + z^2/n)^((n + 1)/2)

F5[z_, n_] := Integrate[f4[u, n], {u, -Infinity, z}]

Plot[{f5[x, 10]}, {x, -10, 10}];

F6[z_, a_, n_] := Integrate[f5[u, n], {u, a, z}]

?F6

F6[z, a, n]

F6[4, a, 6]

Limit[Evaluate[F6[4, a, 6]], a -> -Infinity]

N[%]

Limit[Evaluate[F6[1, a, 6]], a -> -Infinity] // N

Limit[Evaluate[F6[0, a, 6]], a -> -Infinity] // N

Limit[Evaluate[F6[-4, a, 6]], a -> -Infinity] // 
      N

Limit[Evaluate[F6[20, a, 6]], a -> -Infinity] // N

Plot[{F6[x, -1000, 10]}, {x, -10, 10}];

Plot[{F6[x, -10^10, 10]}, {x, -10, 10}];