Lösungen / Statistik 1/08

Remove["Global`*"]

<< Statistics`DescriptiveStatistics`;
<< Statistics`DataManipulation`;
<< Graphics`Graphics`;

1.

a   P(X<=x):=di(x)

d1[x_] := 0;  d2[x_] := x/(2  Pi);  d3[x_] := 1;

b

F[x_] := d1[x] /; x < 0;
F[x_] := d2[x] /; 0 <= x < 2  Pi;
F[x_] := d3[x] /; 2  Pi ≤ x;
Plot[F[x], {x, -2, 8}];

c   f(x)= F'(x)

f[x_] := (Evaluate[D[d1[t], t]] /. t -> x) /; 
          x < 0;
f[x_] := (Evaluate[D[d2[t], t]] /. t -> x) /; 
          0 <= x < 2  Pi;
f[x_] := (Evaluate[D[d3[t], t]] /. t -> x) /; 
        2  Pi ≤ x;  {f[-1], f[2], f[8]}

Plot[f[x], {x, -1, 9}];

2.

a

Remove[f]

Integrate[k * x,  {x, 0, 2}]

s = 
      Solve[Integrate[k * x,  {x, 0, 2}] == 1, {k}] // 
        Flatten

b

f[x_] := 0 /; x < 0;
f[x_] := x/2  /; 0 <= x < 2;
f[x_] := 1 /; 2 ≤ x;
Plot[f[x], {x, -2, 4}];

c

Remove[F]

F[x_] := 0 /; x < 0;
F[x_] := 
        Integrate[t/2, {t, 0, x}]  /; 
          0 <= x < 2;
F[x_] := 1 /; 2 ≤ x;
Plot[F[x], {x, -2, 4}];

3.

a

Remove[f, F, k, x]

Integrate[k * x,  {x, -1, 1}]

k1 = 2;  k1* x

Plot[k1* x, {x, -1, 1}];

Solve[Integrate[k * x,  {x, -1, 1}] == 1, {k}]

Integral=1   nicht erfüllbar / Integral=1   ne pas possible

4.

a

Remove[f, F, k, x]

f1[x_] := 0;
f2[x_] := c  E^(-a x);
Integrate[f2[x], {x, 0, Infinity}]

f2[x_] := Re[c] E^(-Re[a] x);
Integrate[f2[x], {x, 0, Infinity}]

Integrate[f2[x], {x, 0, b}] // Expand

a1 = (Expand[
            Integrate[f2[x], {x, 0, b}]] /. {b Re[a] -> 
              Infinity})

Solve[c/a == 1, {c}]

a0 = 1/2;
f21[x_] := a0  E^(-a0 x);

f[x_] := f1[x] /; x < 0;
f[x_] := f21[x] /; x >= 0;
Plot[f[x], {x, -2, 5}];

Integrate[f21[x], {x, 0, Infinity}]

F[x_] := f1[x] /; x < 0;
F[x_] := 
        Evaluate[Integrate[f21[t], {t, 0, x}]] /; 
          x >= 0;
Plot[F[x], {x, -2, 10}];