# S_Local_Limit_Theorem: Simulations of Local Theorem of DeMoivre-Laplace In ExpRep: Experiment Repetitions

## Description

Given n Bernoulli experiments, with success probability p, this function calculates and plots the exact probability and the approximate probability that a successful event occurs exactly m times (0<=m<=n). It also calculates the difference between these probabilities and shows all the computations in a table.

## Usage

 ```1 2``` ```S_Local_Limit_Theorem(n = 170, p = 0.5, Compare = TRUE, Table = TRUE, Graph = TRUE, GraphE = TRUE) ```

## Arguments

 `n` An integer value representing the number of repetitions of the experiment. `p` A real value with the probability that a successful event will happen in any single Bernoulli experiment (called the probability of success). `Compare` A logical value, if TRUE the function calculates both the exact probability and the approximate probability that a successful event occurs exactly m times and compares these probabilities. `Table` A logical value, if TRUE the function shows a table with the carried out computations. `Graph` A logical value, if TRUE the function plots both the exact probability and the approximate probability that a successful event occurs exactly m times. `GraphE` A logical value, if TRUE the function shows the graphic of the errors in the approximation.

## Details

Bernoulli experiments are sequences of events, in which successive experiments are independent and at each experiment the probability of appearance of a "successful" event (p) remains constant. The value of n must be high and the value of p must be small.

## Value

A graph and/or a table.

## Note

Department of Mathematics. University of Oriente. Cuba.

## Author(s)

Larisa Zamora and Jorge Diaz

## References

Gnedenko, B. V. (1978). The Theory of Probability. Mir Publishers. Moscow.

## See Also

Integral_Theorem, Local_Theorem.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51``` ```S_Local_Limit_Theorem(n = 170, p = 0.5, Compare = TRUE, Table = TRUE, Graph = TRUE, GraphE = TRUE) ## The function is currently defined as function (n = 170, p = 0.5, Compare = TRUE, Table = TRUE, Graph = TRUE, GraphE = TRUE) { layout(matrix(1)) m <- array(0:n) x <- numeric() PNormal <- numeric() a <- n * p b <- sqrt(a * (1 - p)) for (mi in 1:(n + 1)) { x[mi] <- (mi - 1 - a)/b PNormal[mi] <- dnorm(x[mi], 0, 1)/b } if (Compare == TRUE) { PBin <- numeric() for (mi in 1:(n + 1)) PBin[mi] <- dbinom(mi - 1, n, p) Dif <- abs(PBin - PNormal) } if (Graph == TRUE & GraphE == TRUE) { layout(matrix(c(1, 1, 2, 2), 2, 2, byrow = TRUE)) } if (Graph == TRUE) { mfg <- c(1, 1, 2, 2) plot(PNormal, type = "p", main = "The Local Limit Theorem", xlab = "m", ylab = "Probability", col = "red") mtext("Local Theorem", line = -1, side = 3, adj = 1, col = "red") if (Compare == TRUE) { points(PBin, type = "p", col = "blue") mtext("Binomial Probability", line = -2, side = 3, adj = 1, col = "blue") } } if (GraphE == TRUE) { mfg <- c(2, 1, 2, 2) dmini <- min(Dif) - 0.01 dmaxi <- max(Dif) + 0.01 plot(Dif, ylim = c(dmini, dmaxi), type = "b", main = "Errors", xlab = "m", ylab = "Errors", col = "green") abline(a = 0, b = 0, col = "red") } if (Table == TRUE) { if (Compare == TRUE) TablaR <- data.frame(m = m, x = x, PBinomial = PBin, TLocal = PNormal, Difference = Dif) else TablaR <- data.frame(m = m, x = x, TLocal = PNormal) TablaR } } ```

### Example output

```      m          x    PBinomial       TLocal   Difference
1     0 -13.038405 6.681912e-52 7.441914e-39 7.441914e-39
2     1 -12.885012 1.135925e-49 5.434558e-38 5.434558e-38
3     2 -12.731619 9.598566e-48 3.876370e-37 3.876370e-37
4     3 -12.578226 5.375197e-46 2.700644e-36 2.700644e-36
5     4 -12.424833 2.244145e-44 1.837769e-35 1.837769e-35
6     5 -12.271440 7.450561e-43 1.221507e-34 1.221507e-34
7     6 -12.118047 2.048904e-41 7.930160e-34 7.930160e-34
8     7 -11.964654 4.800290e-40 5.028627e-33 5.028626e-33
9     8 -11.811261 9.780591e-39 3.114570e-32 3.114569e-32
10    9 -11.657868 1.760506e-37 1.884205e-31 1.884204e-31
11   10 -11.504475 2.834415e-36 1.113370e-30 1.113367e-30
12   11 -11.351082 4.122786e-35 6.425875e-30 6.425834e-30
13   12 -11.197689 5.462691e-34 3.622482e-29 3.622428e-29
14   13 -11.044296 6.639271e-33 1.994626e-28 1.994560e-28
15   14 -10.890903 7.445468e-32 1.072749e-27 1.072675e-27
16   15 -10.737510 7.743286e-31 5.635287e-27 5.634513e-27
17   16 -10.584117 7.501309e-30 2.891447e-26 2.890697e-26
18   17 -10.430724 6.795303e-29 1.449091e-25 1.448412e-25
19   18 -10.277331 5.776008e-28 7.093452e-25 7.087676e-25
20   19 -10.123938 4.620806e-27 3.391570e-24 3.386949e-24
21   20  -9.970545 3.488709e-26 1.583891e-23 1.580402e-23
22   21  -9.817152 2.491935e-25 7.224885e-23 7.199965e-23
23   22  -9.663759 1.687719e-24 3.218978e-22 3.202101e-22
24   23  -9.510366 1.086011e-23 1.400833e-21 1.389973e-21
25   24  -9.356973 6.651816e-23 5.954372e-21 5.887854e-21
26   25  -9.203580 3.884661e-22 2.472105e-20 2.433259e-20
27   26  -9.050187 2.166445e-21 1.002488e-19 9.808237e-20
28   27  -8.896794 1.155437e-20 3.970752e-19 3.855209e-19
29   28  -8.743401 5.900984e-20 1.536200e-18 1.477190e-18
30   29  -8.590008 2.889448e-19 5.805022e-18 5.516078e-18
31   30  -8.436615 1.358040e-18 2.142601e-17 2.006797e-17
32   31  -8.283222 6.133085e-18 7.724319e-17 7.111010e-17
33   32  -8.129829 2.664059e-17 2.719946e-16 2.453540e-16
34   33  -7.976436 1.114061e-16 9.354956e-16 8.240895e-16
35   34  -7.823043 4.489011e-16 3.142711e-15 2.693810e-15
36   35  -7.669650 1.744301e-15 1.031213e-14 8.567831e-15
37   36  -7.516257 6.541130e-15 3.305018e-14 2.650905e-14
38   37  -7.362864 2.368950e-14 1.034619e-13 7.977241e-14
39   38  -7.209471 8.291324e-14 3.163504e-13 2.334372e-13
40   39  -7.056078 2.806294e-13 9.447952e-13 6.641658e-13
41   40  -6.902685 9.190614e-13 2.756057e-12 1.836996e-12
42   41  -6.749292 2.914097e-12 7.852720e-12 4.938623e-12
43   42  -6.595899 8.950441e-12 2.185412e-11 1.290368e-11
44   43  -6.442506 2.664317e-11 5.940565e-11 3.276248e-11
45   44  -6.289113 7.690188e-11 1.577261e-10 8.082421e-11
46   45  -6.135720 2.153253e-10 4.090352e-10 1.937099e-10
47   46  -5.982327 5.851230e-10 1.036094e-09 4.509707e-10
48   47  -5.828934 1.543729e-09 2.563414e-09 1.019685e-09
49   48  -5.675541 3.955805e-09 6.194693e-09 2.238888e-09
50   49  -5.522148 9.849148e-09 1.462184e-08 4.772694e-09
51   50  -5.368755 2.383494e-08 3.371054e-08 9.875603e-09
52   51  -5.215362 5.608221e-08 7.591204e-08 1.982983e-08
53   52  -5.061969 1.283420e-07 1.669694e-07 3.862743e-08
54   53  -4.908576 2.857425e-07 3.587108e-07 7.296828e-08
55   54  -4.755183 6.191087e-07 7.527197e-07 1.336110e-07
56   55  -4.601790 1.305757e-06 1.542778e-06 2.370214e-07
57   56  -4.448397 2.681465e-06 3.088552e-06 4.070874e-07
58   57  -4.295004 5.362929e-06 6.039315e-06 6.763859e-07
59   58  -4.141611 1.044847e-05 1.153458e-05 1.086114e-06
60   59  -3.988218 1.983437e-05 2.151776e-05 1.683389e-06
61   60  -3.834825 3.669359e-05 3.920792e-05 2.514329e-06
62   61  -3.681432 6.616877e-05 6.978014e-05 3.611363e-06
63   62  -3.528039 1.163290e-04 1.213029e-04 4.973908e-06
64   63  -3.374646 1.994211e-04 2.059642e-04 6.543097e-06
65   64  -3.221253 3.334071e-04 3.415809e-04 8.173802e-06
66   65  -3.067860 5.437101e-04 5.533206e-04 9.610512e-06
67   66  -2.914467 8.649934e-04 8.754702e-04 1.047687e-05
68   67  -2.761074 1.342676e-03 1.352967e-03 1.029078e-05
69   68  -2.607681 2.033760e-03 2.042276e-03 8.516014e-06
70   69  -2.454288 3.006427e-03 3.011084e-03 4.656254e-06
71   70  -2.300895 4.337845e-03 4.336233e-03 1.612615e-06
72   71  -2.147502 6.109641e-03 6.099351e-03 1.029012e-05
73   72  -1.994109 8.400756e-03 8.379845e-03 2.091143e-05
74   73  -1.840716 1.127773e-02 1.124526e-02 3.246377e-05
75   74  -1.687323 1.478297e-02 1.473956e-02 4.340363e-05
76   75  -1.533930 1.892220e-02 1.887039e-02 5.180661e-05
77   76  -1.380537 2.365275e-02 2.359709e-02 5.565483e-05
78   77  -1.227144 2.887478e-02 2.882155e-02 5.322859e-05
79   78  -1.073751 3.442763e-02 3.438410e-02 4.353055e-05
80   79  -0.920358 4.009293e-02 4.006629e-02 2.664315e-05
81   80  -0.766965 4.560571e-02 4.560179e-02 3.915671e-06
82   81  -0.613572 5.067301e-02 5.069511e-02 2.209695e-05
83   82  -0.460179 5.499876e-02 5.504672e-02 4.796844e-05
84   83  -0.306786 5.831193e-02 5.838190e-02 6.996470e-05
85   84  -0.153393 6.039450e-02 6.047923e-02 8.472725e-05
86   85   0.000000 6.110503e-02 6.119495e-02 8.992592e-05
87   86   0.153393 6.039450e-02 6.047923e-02 8.472725e-05
88   87   0.306786 5.831193e-02 5.838190e-02 6.996470e-05
89   88   0.460179 5.499876e-02 5.504672e-02 4.796844e-05
90   89   0.613572 5.067301e-02 5.069511e-02 2.209695e-05
91   90   0.766965 4.560571e-02 4.560179e-02 3.915671e-06
92   91   0.920358 4.009293e-02 4.006629e-02 2.664315e-05
93   92   1.073751 3.442763e-02 3.438410e-02 4.353055e-05
94   93   1.227144 2.887478e-02 2.882155e-02 5.322859e-05
95   94   1.380537 2.365275e-02 2.359709e-02 5.565483e-05
96   95   1.533930 1.892220e-02 1.887039e-02 5.180661e-05
97   96   1.687323 1.478297e-02 1.473956e-02 4.340363e-05
98   97   1.840716 1.127773e-02 1.124526e-02 3.246377e-05
99   98   1.994109 8.400756e-03 8.379845e-03 2.091143e-05
100  99   2.147502 6.109641e-03 6.099351e-03 1.029012e-05
101 100   2.300895 4.337845e-03 4.336233e-03 1.612615e-06
102 101   2.454288 3.006427e-03 3.011084e-03 4.656254e-06
103 102   2.607681 2.033760e-03 2.042276e-03 8.516014e-06
104 103   2.761074 1.342676e-03 1.352967e-03 1.029078e-05
105 104   2.914467 8.649934e-04 8.754702e-04 1.047687e-05
106 105   3.067860 5.437101e-04 5.533206e-04 9.610512e-06
107 106   3.221253 3.334071e-04 3.415809e-04 8.173802e-06
108 107   3.374646 1.994211e-04 2.059642e-04 6.543097e-06
109 108   3.528039 1.163290e-04 1.213029e-04 4.973908e-06
110 109   3.681432 6.616877e-05 6.978014e-05 3.611363e-06
111 110   3.834825 3.669359e-05 3.920792e-05 2.514329e-06
112 111   3.988218 1.983437e-05 2.151776e-05 1.683389e-06
113 112   4.141611 1.044847e-05 1.153458e-05 1.086114e-06
114 113   4.295004 5.362929e-06 6.039315e-06 6.763859e-07
115 114   4.448397 2.681465e-06 3.088552e-06 4.070874e-07
116 115   4.601790 1.305757e-06 1.542778e-06 2.370214e-07
117 116   4.755183 6.191087e-07 7.527197e-07 1.336110e-07
118 117   4.908576 2.857425e-07 3.587108e-07 7.296828e-08
119 118   5.061969 1.283420e-07 1.669694e-07 3.862743e-08
120 119   5.215362 5.608221e-08 7.591204e-08 1.982983e-08
121 120   5.368755 2.383494e-08 3.371054e-08 9.875603e-09
122 121   5.522148 9.849148e-09 1.462184e-08 4.772694e-09
123 122   5.675541 3.955805e-09 6.194693e-09 2.238888e-09
124 123   5.828934 1.543729e-09 2.563414e-09 1.019685e-09
125 124   5.982327 5.851230e-10 1.036094e-09 4.509707e-10
126 125   6.135720 2.153253e-10 4.090352e-10 1.937099e-10
127 126   6.289113 7.690188e-11 1.577261e-10 8.082421e-11
128 127   6.442506 2.664317e-11 5.940565e-11 3.276248e-11
129 128   6.595899 8.950441e-12 2.185412e-11 1.290368e-11
130 129   6.749292 2.914097e-12 7.852720e-12 4.938623e-12
131 130   6.902685 9.190614e-13 2.756057e-12 1.836996e-12
132 131   7.056078 2.806294e-13 9.447952e-13 6.641658e-13
133 132   7.209471 8.291324e-14 3.163504e-13 2.334372e-13
134 133   7.362864 2.368950e-14 1.034619e-13 7.977241e-14
135 134   7.516257 6.541130e-15 3.305018e-14 2.650905e-14
136 135   7.669650 1.744301e-15 1.031213e-14 8.567831e-15
137 136   7.823043 4.489011e-16 3.142711e-15 2.693810e-15
138 137   7.976436 1.114061e-16 9.354956e-16 8.240895e-16
139 138   8.129829 2.664059e-17 2.719946e-16 2.453540e-16
140 139   8.283222 6.133085e-18 7.724319e-17 7.111010e-17
141 140   8.436615 1.358040e-18 2.142601e-17 2.006797e-17
142 141   8.590008 2.889448e-19 5.805022e-18 5.516078e-18
143 142   8.743401 5.900984e-20 1.536200e-18 1.477190e-18
144 143   8.896794 1.155437e-20 3.970752e-19 3.855209e-19
145 144   9.050187 2.166445e-21 1.002488e-19 9.808237e-20
146 145   9.203580 3.884661e-22 2.472105e-20 2.433259e-20
147 146   9.356973 6.651816e-23 5.954372e-21 5.887854e-21
148 147   9.510366 1.086011e-23 1.400833e-21 1.389973e-21
149 148   9.663759 1.687719e-24 3.218978e-22 3.202101e-22
150 149   9.817152 2.491935e-25 7.224885e-23 7.199965e-23
151 150   9.970545 3.488709e-26 1.583891e-23 1.580402e-23
152 151  10.123938 4.620806e-27 3.391570e-24 3.386949e-24
153 152  10.277331 5.776008e-28 7.093452e-25 7.087676e-25
154 153  10.430724 6.795303e-29 1.449091e-25 1.448412e-25
155 154  10.584117 7.501309e-30 2.891447e-26 2.890697e-26
156 155  10.737510 7.743286e-31 5.635287e-27 5.634513e-27
157 156  10.890903 7.445468e-32 1.072749e-27 1.072675e-27
158 157  11.044296 6.639271e-33 1.994626e-28 1.994560e-28
159 158  11.197689 5.462691e-34 3.622482e-29 3.622428e-29
160 159  11.351082 4.122786e-35 6.425875e-30 6.425834e-30
161 160  11.504475 2.834415e-36 1.113370e-30 1.113367e-30
162 161  11.657868 1.760506e-37 1.884205e-31 1.884204e-31
163 162  11.811261 9.780591e-39 3.114570e-32 3.114569e-32
164 163  11.964654 4.800290e-40 5.028627e-33 5.028626e-33
165 164  12.118047 2.048904e-41 7.930160e-34 7.930160e-34
166 165  12.271440 7.450561e-43 1.221507e-34 1.221507e-34
167 166  12.424833 2.244145e-44 1.837769e-35 1.837769e-35
168 167  12.578226 5.375197e-46 2.700644e-36 2.700644e-36
169 168  12.731619 9.598566e-48 3.876370e-37 3.876370e-37
170 169  12.885012 1.135925e-49 5.434558e-38 5.434558e-38
171 170  13.038405 6.681912e-52 7.441914e-39 7.441914e-39
function (n = 170, p = 0.5, Compare = TRUE, Table = TRUE, Graph = TRUE,
GraphE = TRUE)
{
layout(matrix(1))
m <- array(0:n)
x <- numeric()
PNormal <- numeric()
a <- n * p
b <- sqrt(a * (1 - p))
for (mi in 1:(n + 1)) {
x[mi] <- (mi - 1 - a)/b
PNormal[mi] <- dnorm(x[mi], 0, 1)/b
}
if (Compare == TRUE) {
PBin <- numeric()
for (mi in 1:(n + 1)) PBin[mi] <- dbinom(mi - 1, n, p)
Dif <- abs(PBin - PNormal)
}
if (Graph == TRUE & GraphE == TRUE) {
layout(matrix(c(1, 1, 2, 2), 2, 2, byrow = TRUE))
}
if (Graph == TRUE) {
mfg <- c(1, 1, 2, 2)
plot(PNormal, type = "p", main = "The Local Limit Theorem",
xlab = "m", ylab = "Probability", col = "red")
mtext("Local Theorem", line = -1, side = 3, adj = 1,
col = "red")
if (Compare == TRUE) {
points(PBin, type = "p", col = "blue")
mtext("Binomial Probability", line = -2, side = 3,
adj = 1, col = "blue")
}
}
if (GraphE == TRUE) {
mfg <- c(2, 1, 2, 2)
dmini <- min(Dif) - 0.01
dmaxi <- max(Dif) + 0.01
plot(Dif, ylim = c(dmini, dmaxi), type = "b", main = "Errors",
xlab = "m", ylab = "Errors", col = "green")
abline(a = 0, b = 0, col = "red")
}
if (Table == TRUE) {
if (Compare == TRUE)
TablaR <- data.frame(m = m, x = x, PBinomial = PBin,
TLocal = PNormal, Difference = Dif)
else TablaR <- data.frame(m = m, x = x, TLocal = PNormal)
TablaR
}
}
```

ExpRep documentation built on July 4, 2017, 9:45 a.m.