assignment c probability distribution
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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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7/23/2019 Assignment C Probability Distribution
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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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7/23/2019 Assignment C Probability Distribution
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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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7/23/2019 Assignment C Probability Distribution
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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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7/23/2019 Assignment C Probability Distribution
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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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7/23/2019 Assignment C Probability Distribution
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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'(-
+. Poisson Distribution3
N r ' + 4 5 -
- P"$%r
#
.88-4 .44+6 .6-7'8 .'+848 '.-69-x'(4 '.-69-'
x'(5
4. Normal Distribution3
N r ' + 4 5 -
- P"$%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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7/23/2019 Assignment C Probability Distribution
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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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7/23/2019 Assignment C Probability Distribution
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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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7/23/2019 Assignment C Probability Distribution
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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
-
7/23/2019 Assignment C Probability Distribution
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0# )or n = 5 , p=0.5, so q=0.5,
'. Binomial Distribution3
N x ' + 4 5 -
- P"$%x#
.4'+- .'-8+- .4'+- .4'+- .'-8+- .4'+-
+. Poisson Distribution3
N r ' + 4 5 -
- P"$%r
#
.7+7- .+-+' .+-8-+ .+'468 .'448 .887'
4. Normal Distribution3
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
-
7/23/2019 Assignment C Probability Distribution
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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
-
7/23/2019 Assignment C Probability Distribution
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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
-
7/23/2019 Assignment C Probability Distribution
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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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7/23/2019 Assignment C Probability Distribution
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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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