      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
    299
    300
    301
    302
    303
    304
    305
    306
    307
    308
    309
    310
    311
    312

    // Copyright 2018 Developers of the Rand project.
    // Copyright 2016-2017 The Rust Project Developers.
    //
    // Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
    // https://www.apache.org/licenses/LICENSE-2.0> or the MIT license
    // <LICENSE-MIT or https://opensource.org/licenses/MIT>, at your
    // option. This file may not be copied, modified, or distributed
    // except according to those terms.

    //! The binomial distribution.
    #![allow(deprecated)]

    use crate::Rng;
    use crate::distributions::{Distribution, Uniform};

    /// The binomial distribution `Binomial(n, p)`.
    ///
    /// This distribution has density function:
    /// `f(k) = n!/(k! (n-k)!) p^k (1-p)^(n-k)` for `k >= 0`.
    #[deprecated(since="0.7.0", note="moved to rand_distr crate")]
    #[derive(Clone, Copy, Debug)]
    pub struct Binomial {
        /// Number of trials.
        n: u64,
        /// Probability of success.
        p: f64,
    }

    impl Binomial {
        /// Construct a new `Binomial` with the given shape parameters `n` (number
        /// of trials) and `p` (probability of success).
        ///
        /// Panics if `p < 0` or `p > 1`.
        pub fn new(n: u64, p: f64) -> Binomial {
            assert!(p >= 0.0, "Binomial::new called with p < 0");
            assert!(p <= 1.0, "Binomial::new called with p > 1");
            Binomial { n, p }
        }
    }

    /// Convert a `f64` to an `i64`, panicing on overflow.
    // In the future (Rust 1.34), this might be replaced with `TryFrom`.
    fn f64_to_i64(x: f64) -> i64 {
        assert!(x < (::std::i64::MAX as f64));
        x as i64
    }

    impl Distribution<u64> for Binomial {
        fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> u64 {
            // Handle these values directly.
            if self.p == 0.0 {
                return 0;
            } else if self.p == 1.0 {
                return self.n;
            }

            // The binomial distribution is symmetrical with respect to p -> 1-p,
            // k -> n-k switch p so that it is less than 0.5 - this allows for lower
            // expected values we will just invert the result at the end
            let p = if self.p <= 0.5 {
                self.p
            } else {
                1.0 - self.p
            };

            let result;
            let q = 1. - p;

            // For small n * min(p, 1 - p), the BINV algorithm based on the inverse
            // transformation of the binomial distribution is efficient. Otherwise,
            // the BTPE algorithm is used.
            //
            // Voratas Kachitvichyanukul and Bruce W. Schmeiser. 1988. Binomial
            // random variate generation. Commun. ACM 31, 2 (February 1988),
            // 216-222. http://dx.doi.org/10.1145/42372.42381

            // Threshold for prefering the BINV algorithm. The paper suggests 10,
            // Ranlib uses 30, and GSL uses 14.
            const BINV_THRESHOLD: f64 = 10.;

            if (self.n as f64) * p < BINV_THRESHOLD &&
               self.n <= (::std::i32::MAX as u64) {
                // Use the BINV algorithm.
                let s = p / q;
                let a = ((self.n + 1) as f64) * s;
                let mut r = q.powi(self.n as i32);
                let mut u: f64 = rng.gen();
                let mut x = 0;
                while u > r as f64 {
                    u -= r;
                    x += 1;
                    r *= a / (x as f64) - s;
                }
                result = x;
            } else {
                // Use the BTPE algorithm.

                // Threshold for using the squeeze algorithm. This can be freely
                // chosen based on performance. Ranlib and GSL use 20.
                const SQUEEZE_THRESHOLD: i64 = 20;

                // Step 0: Calculate constants as functions of `n` and `p`.
                let n = self.n as f64;
                let np = n * p;
                let npq = np * q;
                let f_m = np + p;
                let m = f64_to_i64(f_m);
                // radius of triangle region, since height=1 also area of region
                let p1 = (2.195 * npq.sqrt() - 4.6 * q).floor() + 0.5;
                // tip of triangle
                let x_m = (m as f64) + 0.5;
                // left edge of triangle
                let x_l = x_m - p1;
                // right edge of triangle
                let x_r = x_m + p1;
                let c = 0.134 + 20.5 / (15.3 + (m as f64));
                // p1 + area of parallelogram region
                let p2 = p1 * (1. + 2. * c);

                fn lambda(a: f64) -> f64 {
                    a * (1. + 0.5 * a)
                }

                let lambda_l = lambda((f_m - x_l) / (f_m - x_l * p));
                let lambda_r = lambda((x_r - f_m) / (x_r * q));
                // p1 + area of left tail
                let p3 = p2 + c / lambda_l;
                // p1 + area of right tail
                let p4 = p3 + c / lambda_r;

                // return value
                let mut y: i64;

                let gen_u = Uniform::new(0., p4);
                let gen_v = Uniform::new(0., 1.);

                loop {
                    // Step 1: Generate `u` for selecting the region. If region 1 is
                    // selected, generate a triangularly distributed variate.
                    let u = gen_u.sample(rng);
                    let mut v = gen_v.sample(rng);
                    if !(u > p1) {
                        y = f64_to_i64(x_m - p1 * v + u);
                        break;
                    }

                    if !(u > p2) {
                        // Step 2: Region 2, parallelograms. Check if region 2 is
                        // used. If so, generate `y`.
                        let x = x_l + (u - p1) / c;
                        v = v * c + 1.0 - (x - x_m).abs() / p1;
                        if v > 1. {
                            continue;
                        } else {
                            y = f64_to_i64(x);
                        }
                    } else if !(u > p3) {
                        // Step 3: Region 3, left exponential tail.
                        y = f64_to_i64(x_l + v.ln() / lambda_l);
                        if y < 0 {
                            continue;
                        } else {
                            v *= (u - p2) * lambda_l;
                        }
                    } else {
                        // Step 4: Region 4, right exponential tail.
                        y = f64_to_i64(x_r - v.ln() / lambda_r);
                        if y > 0 && (y as u64) > self.n {
                            continue;
                        } else {
                            v *= (u - p3) * lambda_r;
                        }
                    }

                    // Step 5: Acceptance/rejection comparison.

                    // Step 5.0: Test for appropriate method of evaluating f(y).
                    let k = (y - m).abs();
                    if !(k > SQUEEZE_THRESHOLD && (k as f64) < 0.5 * npq - 1.) {
                        // Step 5.1: Evaluate f(y) via the recursive relationship. Start the
                        // search from the mode.
                        let s = p / q;
                        let a = s * (n + 1.);
                        let mut f = 1.0;
                        if m < y {
                            let mut i = m;
                            loop {
                                i += 1;
                                f *= a / (i as f64) - s;
                                if i == y {
                                    break;
                                }
                            }
                        } else if m > y {
                            let mut i = y;
                            loop {
                                i += 1;
                                f /= a / (i as f64) - s;
                                if i == m {
                                    break;
                                }
                            }
                        }
                        if v > f {
                            continue;
                        } else {
                            break;
                        }
                    }

                    // Step 5.2: Squeezing. Check the value of ln(v) againts upper and
                    // lower bound of ln(f(y)).
                    let k = k as f64;
                    let rho = (k / npq) * ((k * (k / 3. + 0.625) + 1./6.) / npq + 0.5);
                    let t = -0.5 * k*k / npq;
                    let alpha = v.ln();
                    if alpha < t - rho {
                        break;
                    }
                    if alpha > t + rho {
                        continue;
                    }

                    // Step 5.3: Final acceptance/rejection test.
                    let x1 = (y + 1) as f64;
                    let f1 = (m + 1) as f64;
                    let z = (f64_to_i64(n) + 1 - m) as f64;
                    let w = (f64_to_i64(n) - y + 1) as f64;

                    fn stirling(a: f64) -> f64 {
                        let a2 = a * a;
                        (13860. - (462. - (132. - (99. - 140. / a2) / a2) / a2) / a2) / a / 166320.
                    }

                    if alpha > x_m * (f1 / x1).ln()
                        + (n - (m as f64) + 0.5) * (z / w).ln()
                        + ((y - m) as f64) * (w * p / (x1 * q)).ln()
                        // We use the signs from the GSL implementation, which are
                        // different than the ones in the reference. According to
                        // the GSL authors, the new signs were verified to be
                        // correct by one of the original designers of the
                        // algorithm.
                        + stirling(f1) + stirling(z) - stirling(x1) - stirling(w)
                    {
                        continue;
                    }

                    break;
                }
                assert!(y >= 0);
                result = y as u64;
            }

            // Invert the result for p < 0.5.
            if p != self.p {
                self.n - result
            } else {
                result
            }
        }
    }

    #[cfg(test)]
    mod test {
        use crate::Rng;
        use crate::distributions::Distribution;
        use super::Binomial;

        fn test_binomial_mean_and_variance<R: Rng>(n: u64, p: f64, rng: &mut R) {
            let binomial = Binomial::new(n, p);

            let expected_mean = n as f64 * p;
            let expected_variance = n as f64 * p * (1.0 - p);

            let mut results = [0.0; 1000];
            for i in results.iter_mut() { *i = binomial.sample(rng) as f64; }

            let mean = results.iter().sum::<f64>() / results.len() as f64;
            assert!((mean as f64 - expected_mean).abs() < expected_mean / 50.0,
                    "mean: {}, expected_mean: {}", mean, expected_mean);

            let variance =
                results.iter().map(|x| (x - mean) * (x - mean)).sum::<f64>()
                / results.len() as f64;
            assert!((variance - expected_variance).abs() < expected_variance / 10.0,
                    "variance: {}, expected_variance: {}", variance, expected_variance);
        }

        #[test]
        #[cfg(not(miri))] // Miri is too slow
        fn test_binomial() {
            let mut rng = crate::test::rng(351);
            test_binomial_mean_and_variance(150, 0.1, &mut rng);
            test_binomial_mean_and_variance(70, 0.6, &mut rng);
            test_binomial_mean_and_variance(40, 0.5, &mut rng);
            test_binomial_mean_and_variance(20, 0.7, &mut rng);
            test_binomial_mean_and_variance(20, 0.5, &mut rng);
        }

        #[test]
        fn test_binomial_end_points() {
            let mut rng = crate::test::rng(352);
            assert_eq!(rng.sample(Binomial::new(20, 0.0)), 0);
            assert_eq!(rng.sample(Binomial::new(20, 1.0)), 20);
        }

        #[test]
        #[should_panic]
        fn test_binomial_invalid_lambda_neg() {
            Binomial::new(20, -10.0);
        }
    }

# Help

## Keyboard Shortcuts

`?`
  Show this help dialog
`S`
  Focus the search field
`↑`
  Move up in search results
`↓`
  Move down in search results
`↹`
  Switch tab
`⏎`
  Go to active search result
`+`
  Expand all sections
`-`
  Collapse all sections

## Search Tricks

Prefix searches with a type followed by a colon (e.g., `fn:`) to restrict the search to a given type.

Accepted types are: `fn`, `mod`, `struct`, `enum`, `trait`, `type`, `macro`, and `const`.

Search functions by type signature (e.g., `vec -> usize` or `* -> vec`)

Search multiple things at once by splitting your query with comma (e.g., `str,u8` or `String,struct:Vec,test`)