f @ x_, y_ D := 1 ê H 2 Pi L E ^ H−1 ê 2 H x ^ 2 + y ^ 2 LL ; Plot3D @ f @ x, y D , 8 x, −3, 3 < , 8 y, −3, 3 <D ;

(1)

Lösungen / Statistik 2/08

Remove @ "Global`∗" D

1. ü a

f @ x_, y_ D := 1 ê H 2 Pi L E ^ H−1 ê 2 H x ^ 2 + y ^ 2 LL ; Plot3D @ f @ x, y D , 8 x, −3, 3 < , 8 y, −3, 3 <D ;

-2

0

2 -2 0

2 0

0.05 0.1 0.15

-2

0

2 μX = 3; μY = 2; σX = 3; σY = 2 ê 3; ∑XY = 1 ê 2;

f1 @ x_, y_ D := 1 ê H 2 Pi σX σY Sqrt @ 1 − ∑XY ^ 2 DL E ^ HH −1 ê H 2 H 1 − ∑XY ^ 2 LL

HHH x − μX L ^ 2 L ê H σX ^ 2 L + HH y − μY L ^ 2 L ê H σY ^ 2 L − 2 ∑XY H x − μX L H y − μY L ê H σX σY LLLL ;

? f1 Global`f1

f1 @ x_, y_ D : =

− Hx−μXL2

σX2 +Hy−μYL2

σY2 −2∑XYHx−μXL Hy−μYL σXσY 2H1−∑XY2L 2π σXσYè!!!!!!!!!!!!!!!!!1−∑XY²

(2)

Plot3D @ f1 @ x, y D , 8 x, −2, 8 < , 8 y, −2, 8 < , PlotRange → 8 0, 0.1 <D ;

-2 0

2 4

6 8 -2 0

2 4

6 8

0 0.02 0.04 0.06 0.08 0.1

-2 0

2 4

6 ü b

Integrate @ Integrate @ f @ x, y D , 8 x, −a, a <D , 8 y, −a, a <D Erf A a

è!!! 2 E

²

Integrate @ Integrate @ f @ x, y D , 8 x, −Infinity, Infinity <D , 8 y, −Infinity, Infinity <D 1

Integrate @ Integrate @ f @ x, y D , 8 x, −a, a <D , 8 y, −a, a <D ê . 8 a → Infinity <

1 Integrate @ Integrate @ f @ x, y D , 8 x, −Infinity, Infinity <D , 8 y, −Infinity, Infinity <D 1

Integrate @ Integrate @ f1 @ x, y D , 8 x, −Infinity, Infinity <D , 8 y, −Infinity, Infinity <D 1

Integrate @ f1 @ x, y D , 8 x, −Infinity, Infinity <D 3

⁻⁹⁸ ^H−2+y^L²

2 è!!!!!!! 2 π

Integrate @ f1 @ x, y D , 8 y, −Infinity, Infinity <D

−₁₈¹ H−3+xL²

3 è!!!!!!! 2 π

ü c

F @ x_, y_ D := Integrate @ Integrate @ f @ u, v D , 8 u, −Infinity, x <D , 8 v, −Infinity, y <D

(3)

F @ 0, 0 D 1 4

F @ 0, Infinity D 1

2 F @ Infinity, 0 D 1

2 F @ Infinity, Infinity D 1

F1 @ x_, y_ D := Integrate @ Integrate @ f1 @ u, v D , 8 u, −Infinity, x <D , 8 v, −Infinity, y <D NF1 @ x_, y_ D := NIntegrate @ Integrate @ f1 @ u, v D , 8 u, −Infinity, x <D , 8 v, −Infinity, y <D NF1 @ 3, 2 D

0.333333

2. ü a

Xbar=X1+X2;

m=m1=m2;

s^2=s1^2 / 2 = s2^2 / 2;

ü b

Remove @ f2, F2 D

f2 @ x_, σ_, μ_ D := 1 ê H Sqrt @ 2 Pi D σ L E ^ H −1 ê H 2 L HH x − μ L ^ 2 ê σ^ 2 LL ; F2 @ x_, σ_, μ_ D := Integrate @ f2 @ u, σ, μ D , 8 u, −Infinity, x <D F2 @ Infinity, σ, μD êê Simplify

è!!!!!!! 2 1 π σ If A Re @σ

²

D > 0, è!!!!!!! 2 π è!!!!!! σ

²

, Integrate A

⁻^Hu−μL²^σ²²

, 8 u, −∞, ∞< , Assumptions → Re @σ

²

D ≤ 0 EE

F2 @ Infinity, 4, 5 D 1

F2 @μD

(4)

f3 @ x_, σ_, μ_ D := 1 ê H Sqrt @ 2 Pi D c1 σL E ^ H−1 ê H 2 L HH c1 x + c2 − H c1 μ + c2 LL ^ 2 ê H c1 σL ^ 2 LL ; F3 @ x_, σ_, μ_ D := Integrate @ f3 @ u, σ, μD ∗ Evaluate @ D @ c1 u + c2, u DD , 8 u, −Infinity, x <D F3 @ Infinity, 4, 5 D

1

3. ü a

k @ n_ D := 1 ê H 2 ^ H n ê 2 L Gamma @ n ê 2 DL Table @ k @ n D , 8 n, 0, 10 <D

9 0, 1 è!!!!!!! 2 π , 1

2 , 1 è!!!!!!! 2 π , 1

4 , 1

3 è!!!!!!! 2 π , 1

16 , 1

15 è!!!!!!! 2 π , 1

96 , 1

105 è!!!!!!! 2 π , 1 768 = f4 @ x_, n_ D := k @ n D x ^ HH n − 2 L ê 2 L E ^ H−x ê 2 L

Plot @8 f4 @ x, 10 D , f4 @ x, 5 D< , 8 x, 0, 10 <D ;

2 4 6 8 10

0.025 0.05 0.075 0.1 0.125 0.15

F4 @ x_, n_ D := Integrate @ f4 @ u, n D , 8 u, 0, x <D F4 @ 2, 5 D

− 10

3 è!!! π + Erf @ 1 D F4 @ 2, 5 D êê N 0.150855

ü b

Remove @ f5, f6, F5, F6 D

f5 @ z_, n_ D := Gamma @H n + 1 L ê 2 D ê Sqrt @ n Pi D ê Gamma @ n ê 2 D ê H 1 + z ^ 2 ê n L ^ HH n + 1 L ê 2 L

(5)

Plot @8 f5 @ x, 10 D< , 8 x, −10, 10 <D ;

-10 -5 5 10

0.1 0.2 0.3

F6 @ z_, a_, n_ D := Integrate @ f5 @ u, n D , 8 u, a, z <D

? F6 Global`F6

F6 @ z_, a_, n_ D : = Ÿ

a

z

f5 @ u, n D u

F6 @ z, a, n D i

k jjjjjjH −a + z L Gamma A 1 + n 2 E If Ai

k jjj Im A è!!! n

a − z E + Re A a

a − z E ≥ 1 »» Im A è!!! n

−a + z E + Re A a

−a + z E ≥ 0 »» Im A a − è!!! n a − z E ≠ 0 y

{ zzz &&

i

k jjj Im A è!!! n

−a + z E + Re A a

a − z E ≥ 1 »» Im A è!!! n

−a + z E Re A a

−a + z E »» Im A è!!! n

a − z E ≥ Re A a a − z E »»

Im A a + è!!! n a − z E ≠ 0 y

{ zzz , 1

a − z J a Hypergeometric2F1 A 1

2 , 1 + n 2 , 3

2 , − a

²

n E − z Hypergeometric2F1 A 1

2 , 1 + n 2 , 3

2 , − z

²

n EN , Integrate Ai

k jjj n + H a + u H−a + z LL

²

n

y { zzz

1 2 H−1−nL

, 8 u, 0, 1 < , Assumptions →

! i

k jjji k jjj Im A è!!! n

a − z E + Re A a

a − z E ≥ 1 »» Im A è!!! n

−a + z E + Re A a

−a + z E ≥ 0 »» Im A a − è!!! n a − z E ≠ 0 y

{ zzz &&

i

k jjj Im A è!!! n

−a + z E + Re A a

a − z E ≥ 1 »» Im A è!!! n

−a + z E Re A a

−a + z E »»

Im A è!!! n

a − z E ≥ Re A a

a − z E »» Im A a + è!!! n a − z E ≠ 0 y

{ zzzy { zzzEE y

{ zzzzzz ì Iè!!! n è!!! π Gamma A n 2 EM F6 @ 4, a, 6 D

H 4 − a L I −1127 è!!!!!! 22 +

^{2662 a}^H_H^{135+30 a}_6+a₂_L5ê2²^{+2 a}⁴^L

M 10648 H−4 + a L

Limit @ Evaluate @ F6 @ 4, a, 6 DD , a → −Infinity D 1

2 + 1127 484 è!!!!!! 22 N @ % D

0.996441

(6)

Limit @ Evaluate @ F6 @ 1, a, 6 DD , a → −Infinity D êê N 0.822041

Limit @ Evaluate @ F6 @ 0, a, 6 DD , a → −Infinity D êê N 0.5

Limit @ Evaluate @ F6 @−4, a, 6 DD , a → −Infinity D êê N 0.00355949

Limit @ Evaluate @ F6 @ 20, a, 6 DD , a → −Infinity D êê N 0.999999

Plot @8 F6 @ x, −1000, 10 D< , 8 x, −10, 10 <D ;

-10 -5 5 10

0.2 0.4 0.6 0.8 1

Plot @8 F6 @ x, −10 ^ 10, 10 D< , 8 x, −10, 10 <D ;

-10 -5 5 10

0.005 0.01 0.015 0.02

ü c

F6 @ 2, a, 5 D i

k jjjH 2 − a L i

k jjj −370 + 405 a H 25 + 3 a

²

L

H 5 + a

²

L

²

− 243 è!!! 5 ArcTan A 2

è!!! 5 E + 243 è!!! 5 ArcTan A a è!!! 5 Ey

{ zzzy { zzz ì I 243 è!!! 5 H −2 + a L π M

Limit @ F6 @ 2, a, 5 D , a → −Infinity D 1 +

74è!!!!5

243

+ ArcTan A

è!!!!²5

E

(7)

N @ % D 0.94903

4 Remove @ "Global`∗" D

f @ z_, n_ D := Gamma @H n + 1 L ê 2 D ê H Sqrt @ n Pi D Gamma @ n ê 2 DL ∗ 1 ê H 1 + z ^ 2 ê n L ^ HH n + 1 L ê 2 L ; f @ z, n D

I 1 +

^z_n²

M

¹² ^H−1−n^L

Gamma @

¹⁺ⁿ2

D è!!! n è!!! π Gamma @

ⁿ2

D

F @ z_, a_, n_ D := Gamma @H n + 1 L ê 2 D ê H Sqrt @ n Pi D Gamma @ n ê 2 DL Integrate @ 1 ê H 1 + u ^ 2 ê n L ^ H n + 1 L , 8 u, a, z <D ; F @ z, a, n D i

k jjjjH −a + z L Gamma A 1 + n 2 E If Ai

k jjj Im A è!!! n

a − z E + Re A a

a − z E ≥ 1 »» Im A è!!! n

−a + z E + Re A a

−a + z E ≥ 0 »» Im A a − è!!! n a − z E ≠ 0 y

{ zzz &&

i

k jjj Im A è!!! n

−a + z E + Re A a

a − z E ≥ 1 »» Im A è!!! n

−a + z E Re A a

−a + z E »»

Im A è!!! n

a − z E ≥ Re A a

a − z E »» Im A a + è!!! n a − z E ≠ 0 y

{ zzz , 1 a − z J a Hypergeometric2F1 A 1

2 , 1 + n, 3 2 , − a

²

n E − z Hypergeometric2F1 A 1

2 , 1 + n, 3 2 , − z

²

n EN , Integrate Ai

k jjj n + H a + u H−a + z LL

²

n

y {

zzz

⁻¹⁻ⁿ

, 8 u, 0, 1 < , Assumptions →

! i k jjji

k jjj Im A è!!! n

a − z E + Re A a

a − z E ≥ 1 »» Im A è!!! n

−a + z E + Re A a

−a + z E ≥ 0 »» Im A a − è!!! n a − z E ≠ 0 y

{ zzz &&

i

k jjj Im A è!!! n

−a + z E + Re A a

a − z E ≥ 1 »» Im A è!!! n

−a + z E Re A a

−a + z E »»

Im A è!!! n

a − z E ≥ Re A a

a − z E »» Im A a + è!!! n a − z E ≠ 0 y

{ zzzy { zzzEEy

{ zzzz ì Iè!!! n è!!! π Gamma A n 2 EM F @ z, −Infinity, n D

i k jjjjj

jj Gamma A 1 + n 2 E

If A Re @ n D > − 1 2 ,

"#####

¹

n

è!!! π Gamma @

¹2

+ n D

2 Gamma @ n D + z Hypergeometric2F1 A 1

2 , 1 + n, 3 2 , − z

²

n E , Integrate AJ n + u

²

n N

⁻¹⁻ⁿ

, 8 u, −∞, z < , Assumptions → Re @ n D ≤ − 1 2 EE y

{ zzzzz

zz ì I è!!! n è!!! π Gamma A n

2 EM

(8)

ü a

f @ z, 10 D 63 "#####

⁵

2

256 H 1 +

₁₀^z²

L

¹¹^ê²

Plot @ f @ z, 10 D , 8 z, −5, 5 <D ;

-4 -2 2 4

0.1 0.2 0.3

ü b

F @ z, −Infinity, 10 D i

k jjjjjj$%%%%%% 5 2

i k

jjj 272103300000000000 z + 2909907 è!!!!!! 10 π H 10 + z

²

L

¹⁰

+ 380 z

³

H 402454500000000 + 17 z

²

H 7238280000000 + z

²

H 1396620000000 + 13 z

²

H 13901000000 + 11 z

²

H 111940000 + 6732000 z

²

+ 263760 z

⁴

+ 6090 z

⁶

+ 63 z

⁸

LLLLL +

5819814 è!!!!!! 10 H 10 + z

²

L

¹⁰

ArcTan A z è!!!!!! 10 Ey

{ zzz y

{ zzzzzz ì I 134217728 H 10 + z

²

L

¹⁰

M Plot @ F @ z, −Infinity, 10 D , 8 z, −5, 5 <D ;

-4 -2 2 4

0.1 0.2 0.3 0.4 0.5 0.6

5

(9)

ü a

<< Statistics`DescriptiveStatistics`

data = 8 4.6, 4.5, 4.3, 4.7, 4.5, 4.6, 4.7, 4.5, 4.8 <

8 4.6, 4.5, 4.3, 4.7, 4.5, 4.6, 4.7, 4.5, 4.8 <

Length @ data D 9

locRep = LocationReport @ data D

8 Mean → 4.57778, HarmonicMean → 4.57347, Median → 4.6 <

dispRep = DispersionReport @ data D

8 Variance → 0.0219444, StandardDeviation → 0.148137, SampleRange → 0.5, MeanDeviation → 0.11358, MedianDeviation → 0.1, QuartileDeviation → 0.1 <

ü b

Der Mittelwert der Grundgesamtheit müsste also mit m = 4.5777... und die Standardabweichung der Grundgesamtheit mit s = 0.14813657... eingesetzt werden.

μ = Mean ê . locRep 4.57778

σ = StandardDeviation ê . dispRep 0.148137

ü c

f @ X_ D := Sin @ X − X ^ 2 D ê H 1 − X ^ 2 L − 1 ê X;

f @ X D

− 1

X + Sin @ X − X

²

D 1 − X

²

f @μD

−0.249576

D @ f @ X D , X D 1

X

²

+ H 1 − 2 X L Cos @ X − X

²

D

1 − X

²

+ 2 X Sin @ X − X

²

D H 1 − X

²

L

²

Abs @ D @ f @ X D , X DD Abs @∆X D ê . 8 X → μ, ∆X → σ<

0.0382563

(10)