Assignment C Probability Distribution

7/23/2019 Assignment C Probability Distribution

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Assignment C

Probability Distribution.

Introduction.

In this course work, we are to investigate the approximation of Binomial

distribution in different circumstances by using Poisson and Normal

Distributions.

hese approximations make the calculation less tedious.

he variables, p, the constant probability and , n , the number of trials are

varied.

he probabilities are tabulated using three different distributions for the

purpose of comparison. hese comparisons are also shown graphically. he

investigative study and conclusions are drawn from both the tables and

graphs.

!ethodology.

he formula for Binomial "n,p# is P"$%r# %n r n r

rC p q

where &%'(p .

)or the Poisson "#, P"$%r# %*

re

r

, where %np.

)or the Normal distribution, $N", + #,

P"$%r# % P"r ( .-


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he excel spread sheets are used for the subse&uent calculation where n%'

and n%+ for all the different distributions for p%.-, p%.' and p%.-.

he excel spreadsheets are also used to generate the respective graphs.

0omparisons and investigations on the best method of approximation to

Binomial Distribution where the, n, increased and the , p, varied are then

made.

1esult.

2ssingment 0 "+'-#

2# )or, n = 5, p=0.05, so q=0.95,

'. Binomial Distribution3$

P"$%r# %

N x ' + 4 5 -

- P"$%x

#

.66467 .+484 .+'545 .''+7' +.9877x'(- 4.'+-x'(6

+. Poisson Distribution3

$ % - x .-%.+-

P"$%r# % ,

n r ' + 4 5 -

- P"$%r# .6677 .'956 .+5447 +.+7'x'(4 '.+86-7x'(5 8.4469x'(8

4. Normal Distribution3

$

:%

n r ' + 4 5 -- P"$%r# .845'' .+977+ -.'-66

x'(-'.9598x'(8

'.+98x'(''

'.499-x'('7

;ummary n = 5, p=0.05, so q=0.95

x

Binomi

al Poisson Normal

0

0.7737

81

0.7788

01

0.63411

44


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1

0.2036

27 0.1947

0.29881

99

2

0.0214

34

0.0243

38

0.00515

76

3

0.0011

28

0.0020

28

1.948E-

06

4

2.97E-

05

0.0001

27

1.289E-

11

5

3.13E-

07

6.34E-

06 0

n = 5, p=0.05, so q=0.95

n = 10, p=0.05, so q=0.95

x

Binom

ial

Poisso

n

Norma

l

0

0.000

977

0.006

738

0.0019

61

1

0.009

766

0.033

69

0.0112

15

2

0.043

945

0.084

224

0.0434

95

3

0.117

188

0.140

374

0.1144

68

4

0.205

078

0.175

467

0.2045

24

5

0.246

094

0.175

467

0.2481

7


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6

0.205

078

0.146

223

0.2045

24

7

0.117

188

0.104

445

0.1144

68

8

0.043

945

0.065

278

0.0434

95

9

0.009

766

0.036

266

0.0112

15

10

0.000

977

0.018

133

0.0019

61

n = 10, p=0.05, so q=0.95

n = 20, p=0.05, so q=0.95

x

Binom

ial

Poisso

n

Norma

l

0

9.54E-

07

4.54E-

05

9.43E-

06

1

1.91E-

05

0.000

454

6.12E-

05

2

0.000

181

0.002

27

0.0003

26

3 0.001 0.007 0.0014


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087 567 27

4

0.004

621

0.018

917

0.0051

28

5

0.014

786

0.037

833

0.0151

33

6

0.036

964

0.063

055

0.0366

77

7

0.073

929

0.090

079

0.0730

14

8

0.120

134

0.112

599

0.1193

91

9

0.160

179

0.125

11

0.1603

64

10

0.176

197

0.125

11

0.1769

37

11

0.160

179

0.113

736

0.1603

64

120.120

1340.094

780.1193

91

13

0.073

929

0.072

908

0.0730

14

14

0.036

964

0.052

077

0.0366

77

15

0.014

786

0.034

718

0.0151

33

16

0.004

621

0.021

699

0.0051

28

17

0.001

087

0.012

764

0.0014

27

180.000

1810.007

0910.0003

26

19

1.91E-

05

0.003

732

6.12E-

05

20

9.54E-

07

0.001

866

9.43E-

06

n = 20, p=0.05, so q=0.95


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B# )or, n = 5, p=0.1 , so q=0.9,'. Binomial Distribution3

N x ' + 4 5 -

- P"$%x

#

.-959 .4+7- .6+9 7.'x'(4 5.-x'(5 'x'(-


N r ' + 4 5 -

- P"$%r

#

.88-4 .44+6 .6-7'8 .'+848 '.-69-x'(4 '.-69-'

x'(5


N r ' + 4 5 -

- P"$%r

#

.54'97 .54'97 .88-75 '.546

x'(44.7655x'(8 '.+546x'(9

;ummary n = 5, p=0.1 , so q=0.9

x

Binomi

al Poisson Normal

0

0.5904

9

0.6065

31

0.43198

14

1

0.3280

5

0.3032

65

0.43198

14

2 0.07290.0758

160.06658

4

3 0.0081

0.0126

36

0.00143

07

4

0.0004

5

0.0015

8

3.871E-

06

5

0.0000

1

0.0001

58

1.239E-

09


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n = 5, p=0.1 , so q=0.9

n = 10, p=0.1 , so q=0.9

x

Binom

ial

Poisso

n

Norma

l

00.348

6780.367

8790.2421

58

1

0.387

42

0.367

879

0.4018

39

2

0.193

71

0.183

94

0.2421

58

3

0.057

396

0.061

313

0.0527

19

4

0.011

16

0.015

328

0.0040

92

5

0.001

488

0.003

066

0.0001

11

6

0.000

138

0.000

511

1.05E-

06

7

8.75E-

06

7.3E-

05

3.36E-

09

8

3.65E-

07

9.12E-

06

3.65E-

12

9 9E-09

1.01E-

06 0


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10 1E-10

1.01E-

07 0

n = 10, p=0.1 , so q=0.9

n = 20, p=0.1 , so q=0.9

x

Binom

ial

Poisso

n

Norma

l

0

0.121

577

0.135

335

0.1005

73

1

0.270

17

0.270

671

0.2229

18

2

0.285

18

0.270

671

0.2906

12

3

0.190

12

0.180

447

0.2229

18

4 0.089 0.090 0.1005


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779 224 73

5

0.031

921

0.036

089

0.0266

6

6

0.008

867

0.012

03

0.0041

46

7

0.001

97

0.003

437

0.0003

77

8

0.000

356

0.000

859

2.01E-

05

9

5.27E-

05

0.000

191

6.22E-

07

10

6.44E-

06

3.82E-

05

1.12E-

08

11

6.51E-

07

6.94E-

06

1.18E-

10

12

5.42E-

08

1.16E-

06

7.14E-

13

133.71E-

091.78E-

072.55E-

15

14

2.06E-

10

2.54E-

08 0

15

9.15E-

12

3.39E-

09 0

16

3.18E-

13

4.24E-

10 0

17

8.31E-

15

4.99E-

11 0

18

1.54E-

16

5.54E-

12 0

191.8E-

185.83E-

13 0

20 1E-20

5.83E-

14 0

n = 20, p=0.1 , so q=0.9


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0# )or n = 5 , p=0.5, so q=0.5,

'. Binomial Distribution3

N x ' + 4 5 -

- P"$%x#

.4'+- .'-8+- .4'+- .4'+- .'-8+- .4'+-


N r ' + 4 5 -

- P"$%r

#

.7+7- .+-+' .+-8-+ .+'468 .'448 .887'


N r ' + 4 5 -

- P"$%r #

.44'65 .'5764 .4'55- .4'55- .'5764 .44'65

;ummary n = 5 , p=0.5, so q=0.5

x

Binomi

al Poisson Normal

0

0.0312

5

0.0820

85

0.03317

4

1

0.1562

5

0.2052

12

0.14872

75

2 0.3125

0.2565

16

0.31445

33

3 0.3125

0.2137

63

0.31445

33

4

0.1562

5

0.1336

02

0.14872

75


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5

0.0312

5

0.0668

01

0.03317

4

n = 5, p=0.5, so q=0.5

n = 10, p=0.5, so q=0.5

x

Binom

ial

Poisso

n

Norma

l

0

0.000

977

0.006

738

0.0019

61

1

0.009

766

0.033

69

0.0112

15

2

0.043

945

0.084

224

0.0434

95

3

0.117

188

0.140

374

0.1144

68

4

0.205

078

0.175

467

0.2045

24

5

0.246

094

0.175

467

0.2481

7

6

0.205

078

0.146

223

0.2045

24

7

0.117

188

0.104

445

0.1144

68

8

0.043

945

0.065

278

0.0434

95


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9

0.009

766

0.036

266

0.0112

15

10

0.000

977

0.018

133

0.0019

61

n = 10, p=0.5, so q=0.5

n = 20, p=0.5, so q=0.5

x

Binom

ial

Poisso

n

Norma

l

0

9.54E-

07

4.54E-

05

9.43E-

06

1

1.91E-

05

0.000

454

6.12E-

05

2

0.000

181

0.002

27

0.0003

26

3

0.001

087

0.007

567

0.0014

27

4

0.004

621

0.018

917

0.0051

28


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5

0.014

786

0.037

833

0.0151

33

6

0.036

964

0.063

055

0.0366

77

7

0.073

929

0.090

079

0.0730

14

8

0.120

134

0.112

599

0.1193

91

9

0.160

179

0.125

11

0.1603

64

10

0.176

197

0.125

11

0.1769

37

11

0.160

179

0.113

736

0.1603

64

12

0.120

134

0.094

78

0.1193

91

13

0.073

929

0.072

908

0.0730

14

14

0.036

964

0.052

077

0.0366

77

15

0.014

786

0.034

718

0.0151

33

16

0.004

621

0.021

699

0.0051

28

17

0.001

087

0.012

764

0.0014

27

18

0.000

181

0.007

091

0.0003

26

19

1.91E-

05

0.003

732

6.12E-

05

20

9.54E-

07

0.001

866

9.43E-

06

n = 20, p=0.5, so q=0.5


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0onclusion.

)rom the results, it can be seen that 3

i# the Poisson distribution is a good approximation to the Binomial

distribution when the, p , is small and, n, is big.

ii# he Normal distribution is a good approximation to the Binomial

distribution when p is .- and the, n is big.

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Assignment C Probability Distribution