Lösungen / Statistik 2/03

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Lösungen / Statistik 2/03

Remove @ "Global`∗" D

1. f @ ν_, μ_, σ_ D := 1 ê H σ Sqrt @ 2 Pi DL E ^ H −1 ê 2 ∗ H ν − μ L ^ 2 ê σ ^ 2 L ;

F @ x_, μ_, σ_ D := Evaluate @ Integrate @ f @ ν, μ, σ D , 8 ν, −Infinity, x <DD

ü a

Plot @ f @ν, 0, 1 D , 8ν, −5, 5 <D ;

-4 -2 2 4

0.1 0.2 0.3 0.4

Plot @ F @ x, 0, 1 D , 8 x, −5, 5 <D ;

-4 -2 2 4

0.2

0.4

0.6

0.8

1

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Plot @8 f @ x, 0, 1 D , F @ x, 0, 1 D< , 8 x, −5, 5 <D ;

-4 -2 2 4

0.2 0.4 0.6 0.8 1

ü b

Plot @8 f @ x, 0, 2 D , F @ x, 0, 2 D< , 8 x, −5, 5 <D ;

-4 -2 2 4

0.2 0.4 0.6 0.8 1

ü c

Plot @8 f @ x, 0, 1 ê 2 D , F @ x, 0, 1 ê 2 D< , 8 x, −5, 5 <D ;

-4 -2 2 4

0.2

0.4

0.6

0.8

1

(3)

ü d

Plot @8 f @ x, 3, 1 D , F @ x, 3, 1 D< , 8 x, −5, 8 < , PlotRange → 8 0, 1 <D ;

-4 -2 2 4 6 8

0.2 0.4 0.6 0.8 1

ü d

Plot @8 f @ x, 3, 2 D , F @ x, 3, 2 D< , 8 x, −5, 8 < , PlotRange → 8 0, 1 <D ;

-4 -2 2 4 6 8

0.2 0.4 0.6 0.8 1

ü e

Plot @8 f @ x, 3, 1 ê 2 D , F @ x, 3, 1 ê 2 D< , 8 x, −5, 8 < , PlotRange → 8 0, 1 <D ;

-4 -2 2 4 6 8

0.2

0.4

0.6

0.8

1

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2. Plot @8 f @ x, 30, 0.02 D , F @ x, 30, 0.02 D< , 8 x, 29.8, 30.2 < , PlotRange → 8 0, 20 <D ;

29.8 29.9 30.1 30.2

2.5 5 7.5 10 12.5 15 17.5 20

Plot @8 F @ x, 30, 0.02 D< , 8 x, 29.9, 30.1 <D ;

29.9 29.95 30.05 30.1

0.2 0.4 0.6 0.8 1

Ausschuss / Rebut ( % ) :

HH 1 − F @ 30 + 0.03, 30, 0.02 DL + F @ 30 − 0.03, 30, 0.02 DL 0.133614

HH 1 − F @ 30 + 0.03, 30, 0.02 DL + F @ 30 − 0.03, 30, 0.02 DL ∗ 100 êê N 13.3614

3. **Remove["Global`*"]**

ü a

n = 400; p = 0.07; m = 10;

<< Statistics`DiscreteDistributions`

bdist @ n_ D := BinomialDistribution @ n, p D

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pdf @ n_, x_ D := PDF @ bdist @ n D , x D ; cdf @ n_, m_ D := CDF @ bdist @ n D , m D ; 1 − cdf @ n, m − 1 D

0.999982

Sum @ pdf @ n, x D , 8 x, m, n <D 0.999982

ü b

f @ν_, μ_, σ_ D := 1 ê Hσ Sqrt @ 2 Pi DL E ^ H−1 ê 2 ∗ Hν − μL ^ 2 ê σ ^ 2 L ;

F @ x_, μ_, σ_ D := Evaluate @ Integrate @ f @ν, μ, σD , 8ν, −Infinity, x <DD q = 1 − p; μ = n ∗ p; σ = Sqrt @μ ∗ q D ;

1 − F @ 10, μ, σD êê N 0.99979

1 − F @ 9, μ, σD êê N 0.999902

1 − F @ 9.5, μ, σ D êê N 0.999856

4. **Remove["Global`*"]**

points =

88 0, 0.497 < , 8 1, 0.580 < , 8 2, 0.839 < , 8 3, 0.933 < , 8 4, 1.044 < , 8 5, 1.141 < , 8 6, 1.151 < , 8 7, 1.313 < , 8 8, 1.404 < , 8 9, 1.409 < , 8 10, 1.422 < , 8 11, 1.451 < , 8 12, 1.529 << ;

ü a

f1 = FindFit @ points, a + b x, 8 a, b < , x D 8 a → 0.630703, b → 0.083511 <

fg1 @ x_ D := a + b x ê . f1; fg1 @ x D

0.630703 + 0.083511 x

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plotpoints = ListPlot @ points, PlotStyle → PointSize @ 0.02 DD ;

2 4 6 8 10 12

0.6 0.8 1.2 1.4

pl1 = Plot @ fg1 @ x D , 8 x, 0, 12 <D ;

2 4 6 8 10 12

0.8 1.2 1.4 1.6

Show @ pl1, plotpoints D ;

2 4 6 8 10 12

0.6 0.8 1.2 1.4 1.6

ü b

f2 = FindFit @ points, a + b x + c x ^ 2, 8 a, b, c < , x D 8 a → 0.490363, b → 0.16006, c → −0.00637912 <

fg2 @ x_ D := a + b x + c x ^ 2 ê . f2; fg2 @ x D

0.490363 + 0.16006 x − 0.00637912 x

²

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pl2 = Plot @ fg2 @ x D , 8 x, 0, 12 <D ;

2 4 6 8 10 12

0.6 0.8 1.2 1.4

Show @ pl1, pl2, plotpoints D ;

2 4 6 8 10 12

0.6 0.8 1.2 1.4 1.6

ü c

sin = a + b Sin @ c + d x D ;

f3 = FindFit @ points, sin, 8 a, b, c, d < , x D ; sin ê . f3

1.13073 − 0.179832 Sin @ 0.84797 + 0.929293 x D

Plot @ sin ê . f3, 8 x, −1, 12 < , AxesOrigin → 8 0, 0 < , PlotRange → 8 Automatic, 8 0, 2 << , Epilog → Prepend @ Point ê@ points, PointSize @ 0.02 DDD ;

2 4 6 8 10 12

0.25

0.5

0.75

1

1.25

1.5

1.75

2

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ü d

trigpoly = c + d Cos @ x D + e Cos @ x D

²

+ f Cos @ x D

³

+ g Sin @ x D + h Sin @ x D

²

+ i Sin @ x D

³

; f4 = FindFit @ points, trigpoly, 8 c, d, e, f, g, h, i < , x D

8 c → 0.487792, d → 0.231707, e → 0.572626,

f → −0.427833, g → −0.448662, h → 0.722827, i → 0.398751 <

trigpoly ê . f4

0.487792 + 0.231707 Cos @ x D + 0.572626 Cos @ x D

²

−

0.427833 Cos @ x D

³

− 0.448662 Sin @ x D + 0.722827 Sin @ x D

²

+ 0.398751 Sin @ x D

³

Plot @ trigpoly ê . f4, 8 x, −1, 12 < , AxesOrigin → 8 0, 0 < ,

PlotRange → 8 Automatic, 8 0, 2 << , Epilog → Prepend @ Point ê @ points, PointSize @ 0.02 DDD ;

2 4 6 8 10 12

0.25 0.5 0.75 1 1.25 1.5 1.75 2

ü e

? Fit

System`

FindFit Fit FitAll