Lösungen / Statistik 1/08

Lösungen / Statistik 1/08

Remove @ "Global`*" D

<< Statistics`DescriptiveStatistics`;

<< Statistics`DataManipulation`;

<< Graphics`Graphics`;

1.

Ÿ a P(X<=x):=di(x)

d1 @ x_ D := 0; d2 @ x_ D := x • H 2 Pi L ; d3 @ x_ D := 1;

Ÿ b

F @ x_ D := d1 @ x D • ; x < 0;

F @ x_ D := d2 @ x D • ; 0 <= x < 2 Pi;

F @ x_ D := d3 @ x D • ; 2 Pi £ x;

Plot @ F @ x D , 8 x, -2, 8 <D ;

-2 -1 1 2

0.2 0.4 0.6 0.8 1

Ÿ c f(x)= F'(x)

Plot @ f @ x D , 8 x, -1, 9 <D ;

2 4 6 8

0.025 0.05 0.075 0.1 0.125 0.15

2.

Ÿ a

Remove @ f D

Integrate @ k * x, 8 x, 0, 2 <D 2 k

s = Solve @ Integrate @ k * x, 8 x, 0, 2 <D Š 1, 8 k <D •• Flatten 9 k ® 1

€€€€ 2 =

Ÿ b

f @ x_ D := 0 • ; x < 0;

f @ x_ D := x • 2 • ; 0 <= x < 2;

f @ x_ D := 1 • ; 2 £ x;

Plot @ f @ x D , 8 x, -2, 4 <D ;

-2 -1 1 2

0.2

0.4

0.6

0.8

1

Ÿ c

Remove @ F D

F @ x_ D := 0 • ; x < 0;

F @ x_ D := Integrate @ t • 2, 8 t, 0, x <D • ; 0 <= x < 2;

F @ x_ D := 1 • ; 2 £ x;

Plot @ F @ x D , 8 x, -2, 4 <D ;

-2 -1 1 2

0.2 0.4 0.6 0.8 1

3.

Ÿ a

Remove @ f, F, k, x D

Integrate @ k * x, 8 x, -1, 1 <D 0

k1 = 2; k1 * x 2 x

Plot @ k1 * x, 8 x, -1, 1 <D ;

1

2

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

4.

Ÿ a

Remove @ f, F, k, x D f1 @ x_ D := 0;

f2 @ x_ D := c E ^ H-a x L ;

Integrate @ f2 @ x D , 8 x, 0, Infinity <D c If A Re @ a D > 0, 1

€€€€ a , Integrate @ã

, 8 x, 0, ¥< , Assumptions ® Re @ a D £ 0 DE

f2 @ x_ D := Re @ c D E ^ H-Re @ a D x L ; Integrate @ f2 @ x D , 8 x, 0, Infinity <D If A Re @ a D > 0, 1

€€€€€€€€€€€€€€€€

Re @ a D , Integrate @ ã

, 8 x, 0, ¥ < , Assumptions ® Re @ a D £ 0 DE Re @ c D

Integrate @ f2 @ x D , 8 x, 0, b <D •• Expand Re @ c D

€€€€€€€€€€€€€€€€

Re @ a D - €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ ã

Re @ c D Re @ a D

a1 = H Expand @ Integrate @ f2 @ x D , 8 x, 0, b <DD • . 8 b Re @ a D ® Infinity <L Re @ c D

€€€€€€€€€€€€€€€€

Re @ a D

Solve @ c • a == 1, 8 c <D 88 c ® a <<

a0 = 1 • 2;

f21 @ x_ D := a0 E ^ H-a0 x L ; f @ x_ D := f1 @ x D • ; x < 0;

f @ x_ D := f21 @ x D • ; x >= 0;

Plot @ f @ x D , 8 x, -2, 5 <D ;

-2 -1 1 2

0.1

0.2

0.3

0.4

0.5

Integrate @ f21 @ x D , 8 x, 0, Infinity <D 1

F @ x_ D := f1 @ x D • ; x < 0;

F @ x_ D := Evaluate @ Integrate @ f21 @ t D , 8 t, 0, x <DD • ; x >= 0;

Plot @ F @ x D , 8 x, -2, 10 <D ;

-2 -1 1 2

0.2

0.4

0.6

0.8

1