-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathstatslib_docs_binom.html
298 lines (213 loc) · 7.62 KB
/
statslib_docs_binom.html
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
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="description" content="StatsLib is a templated C++ library for fast computation of statistical distribution functions.">
<meta name="author" content="Keith O'Hara">
<meta name="keywords" content="Statistics, Probability, Distributions, C++, Cpp, C++11, constexpr, Econometrics" />
<link rel="shortcut icon" type="image/x-icon" href="siteicon.ico">
<title>StatsLib: Binomial Distribution</title>
<!-- Bootstrap Core CSS -->
<link href="css/bootstrap.min.css" rel="stylesheet">
<!-- Custom CSS -->
<link href="css/modern-business.css" rel="stylesheet">
<!-- Custom Fonts -->
<link href="font-awesome/css/font-awesome.min.css" rel="stylesheet" type="text/css">
<!-- Additional Settings -->
<link href="css/yuxuan_settings.css" rel="stylesheet">
<!-- Syntax Highlighter -->
<script type="text/javascript" src="js/syntaxhighlighter.js"></script>
<link type="text/css" rel="stylesheet" href="css/swift_theme.css">
<script>
(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
})(window,document,'script','https://www.google-analytics.com/analytics.js','ga');
ga('create', 'UA-93902857-1', 'auto');
ga('send', 'pageview');
</script>
<!-- MathJax -->
<script type="text/x-mathjax-config">
MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}});
</script>
<script type="text/javascript" async
src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_CHTML">
</script>
<script async defer src="https://buttons.github.io/buttons.js"></script>
<script src="js/jquery.js"></script>
<script>
$(function(){
$("#mynavbar").load("navbar.html")
$("#statshead").load("statslib_header.html")
$("#myfooter").load("footer.html")
});
</script>
</head>
<style>
pre {
display: inline-block;
}
</style>
<body>
<!-- Navigation -->
<div id="mynavbar"></div>
<!-- Page Content -->
<div class="container">
<!-- Page Heading/Breadcrumbs -->
<div id="statshead"></div>
<!-- -->
<br>
<div class="row">
<div class="col-md-2"></div>
<div class="col-md-5">
<div class="panel panel-default">
<div class="panel-heading">
<a data-toggle="collapse" href="#collapse1"><h4><strong style="font-size: 120%;">StatsLib: Binomial Distribution</strong></h4></a>
</div>
<div id="collapse1" class="panel-collapse collapse">
<div class="panel-body">
<a href="#intro">Introduction</a> <br>
<a href="#density">Density</a> <br>
<a href="#probability">Probability</a> <br>
<a href="#quantile">Quantile</a> <br>
<a href="#random">Random Sampling</a>
</div>
</div>
</div>
</div>
</div>
<!-- -->
<div class="row">
<div class="col-md-2"></div>
<div class="col-md-8">
<hr style="height:2px;border-width:0;background-color:black">
<h3 style="text-align: left;" id="intro"><strong style="font-size: 120%;">Introduction</strong></h3>
<p>The density function of the Binomial distribution:</p>
$$
f(x; n, p) = \binom{n}{x} p^x (1-p)^{n-x}
$$
<p>Parameters:</p>
<ul>
<li><code>n_trials_par</code> is $n$</li>
<li><code>prob_par</code> is $p$</li>
</ul>
<hr style="height:2px;border-width:0;background-color:black">
<!-- -->
<h3 style="text-align: left;" id="density"><strong style="font-size: 120%;">Density</strong></h3>
<br>
<p><strong>Definition:</strong></p>
<pre class="brush: cpp;">
template<typename T>
statslib_constexpr
T dbinom(const uint_t x, const uint_t n_trials_par, const T prob_par, const bool log_form = false);
</pre>
<p>Computes the density function.</p>
<hr>
<p><strong>Examples:</strong></p>
<pre class="brush: cpp;">
// parameters
double prob_par = 0.6;
unsigned int n_trials = 4;
// standard input
double dens_val = stats::dbinom(2,n_trials,prob_par);
double log_dens_val = stats::dbinom(2,n_trials,prob_par,true);
// Armadillo input
arma::mat X(10,1);
X.fill(1);
arma::mat dens_vals_mat = stats::dbinom(X,n_trials,prob_par);
arma::mat log_dens_vals_mat = stats::dbinom(X,n_trials,prob_par,true);
</pre>
<hr style="height:2px;border-width:0;background-color:black">
<!-- -->
<h3 style="text-align: left;" id="probability"><strong style="font-size: 120%;">Probability</strong></h3>
<br>
<p><strong>Definition:</strong></p>
<pre class="brush: cpp;">
template<typename T>
statslib_constexpr
T pbinom(const uint_t x, const uint_t n_trials_par, const T prob_par, const bool log_form = false);
</pre>
<p>Computes the cumulative distribution function (CDF).</p>
<hr>
<p><strong>Examples:</strong></p>
<pre class="brush: cpp;">
// parameters
double prob_par = 0.6;
unsigned int n_trials = 4;
// standard input
double prob_val = stats::pbinom(2,n_trials,prob_par);
double log_prob_val = stats::pnorm(2,n_trials,prob_par,true);
// Armadillo input
arma::mat X(10,1);
X.fill(2);
arma::mat prob_vals_mat = stats::pbinom(X,n_trials,prob_par);
arma::mat log_prob_vals_mat = stats::pbinom(X,n_trials,prob_par,true);
</pre>
<hr style="height:2px;border-width:0;background-color:black">
<!-- -->
<h3 style="text-align: left;" id="quantile"><strong style="font-size: 120%;">Quantile</strong></h3>
<br>
<p><strong>Definition:</strong></p>
<pre class="brush: cpp;">
template<typename Ta, typename Tb = uint_t>
statslib_constexpr
Tb qbinom(const Ta p, const uint_t n_trials_par, const Ta prob_par);
</pre>
<p>Computes the quantile function.</p>
<hr>
<p><strong>Examples:</strong></p>
<pre class="brush: cpp;">
// parameters
double prob_par = 0.6;
unsigned int n_trials = 4;
// standard input
int quant_val = stats::qbinom(0.7,n_trials,prob_par);
// Armadillo input
arma::mat X(10,1);
X.fill(0.7);
arma::mat quant_vals_mat = stats::qbinom(X,n_trials,prob_par);
</pre>
<hr style="height:2px;border-width:0;background-color:black">
<!-- -->
<h3 style="text-align: left;" id="random"><strong style="font-size: 120%;">Random Sampling</strong></h3>
<br>
<p><strong>Definition:</strong></p>
<pre class="brush: cpp;">
// random engine seeding
template<typename T>
statslib_inline
uint_t rbinom(const uint_t n_trials_par, const T prob_par, rand_engine_t& engine);
// seeding values
template<typename T>
sstatslib_inline
uint_t rbinom(const uint_t n_trials_par, const T prob_par, uint_t seed_val = std::random_device{}());
// matrix output
template<typename mT, typename eT>
statslib_inline
mT rbinom(const uint_t n, const uint_t k, const uint_t n_trials_par, const eT prob_par);
</pre>
<p>Generates pseudo-random draws.</p>
<hr>
<p><strong>Examples:</strong></p>
<pre class="brush: cpp;">
// parameters
double prob_par = 0.6;
unsigned int n_trials = 4;
// standard input
int rand_val = stats::rbinom(n_trials,prob_par);
// Armadillo output: 10 x 1 matrix
arma::mat rand_mat = stats::rbinom<arma::mat>(10,1,n_trials,prob_par);
</pre>
<hr style="height:2px;border-width:0;background-color:black">
</div>
</div>
</div>
<div id="myfooter"></div>
<!-- jQuery -->
<!--<script src="js/jquery.js"></script>-->
<!-- Bootstrap Core JavaScript -->
<script src="js/bootstrap.min.js"></script>
</body>
</html>