Binomial Distribution

Table of contents

Density Function

The density function of the Binomial distribution:

\[f(x; n, p) = \binom{n}{x} p^x (1-p)^{n-x} \times \mathbf{1}[x \in \{0,\ldots,n\}]\]

Methods for scalar input, as well as for list input, are listed below.

Scalar Input

pystats.dbinom(x: float, n_trials: int, prob: float, log: bool = False) → float

Density function of the Binomial distribution.

Example

>>> pystats.dbinom(2, 4, 0.4)
0.3456
Parameters

  • x (int) – An integral-valued input, equal to 0 or 1.

  • n_trials (int) – The number of trials, a non-negative integral-valued input.

  • prob (float) – The probability parameter, a real-valued input.

  • log (bool) – Return the log-density or the true form.

Returns

The density function evaluated at x.

List Input

pystats.dbinom(x: List[float], n_trials: int, prob: float, log: bool = False) → List[float]

Density function of the Binomial distribution.

Example

>>> pystats.dbinom([2, 3, 4], 5, 0.4)
[0.3456, 0.2304, 0.0768]
Parameters

  • x (List[int]) – A standard list input.

  • n_trials (int) – The number of trials, a non-negative integral-valued input.

  • prob (float) – The probability parameter, a real-valued input.

  • log (bool) – Return the log-density or the true form.

Returns

A list of density values corresponding to the elements of x.

Cumulative Distribution Function

The cumulative distribution function (CDF) of the Binomial distribution:

\[F(x; n, p) = \sum_{z \leq x} f(z; n, p)\]

Methods for scalar input, as well as for list input, are listed below.

Scalar Input

pystats.pbinom(p: float, n_trials: int, prob: float, log: bool = False) → float

Distribution function of the Binomial distribution.

Example

>>> pystats.pbinom(2, 4, 0.4)
0.8208
Parameters

  • p (float) – A value equal to 0 or 1.

  • n_trials (int) – The number of trials, a non-negative integral-valued input.

  • prob (float) – The probability parameter, a real-valued input.

  • log (bool) – Return the log-density or the true form.

Returns

The cumulative distribution function evaluated at p.

List Input

pystats.pbinom(p: List[float], n_trials: int, prob: float, log: bool = False) → List[float]

Distribution function of the Binomial distribution.

Example

>>> pystats.pbinom([2, 3, 4], 5, 0.4)
[0.68256, 0.91296, 0.98976]
Parameters

  • p (List[float]) – A standard list input.

  • n_trials (int) – The number of trials, a non-negative integral-valued input.

  • prob (float) – The probability parameter, a real-valued input.

  • log (bool) – Return the log-density or the true form.

Returns

A list of CDF values corresponding to the elements of p.

Quantile Function

The quantile function of the Binomial distribution:

\[q(r; n, p) = \inf \left\{ x : r \leq F(x; n, p) \right\}\]

Methods for scalar input, as well as for list input, are listed below.

Scalar Input

pystats.qbinom(q: float, n_trials: int, prob: float) → float

Quantile function of the Binomial distribution.

Example

>>> pystats.qbinom(0.5, 4, 0.4)
2
Parameters

  • q (float) – A real-valued input.

  • n_trials (int) – The number of trials, a non-negative integral-valued input.

  • prob (float) – The probability parameter, a real-valued input.

Returns

The quantile function evaluated at q.

List Input

pystats.qbinom(q: List[float], n_trials: int, prob: float) → List[float]

Quantile function of the Binomial distribution.

Example

>>> pystats.qbinom([0.2, 0.4, 0.8], 5, 0.4)
[1, 2, 3]
Parameters

  • q (List[float]) – A standard list input.

  • n_trials (int) – The number of trials, a non-negative integral-valued input.

  • prob (float) – The probability parameter, a real-valued input.

Returns

A list of quantiles values corresponding to the elements of q.

Random Sampling

Random sampling for the Binomial distribution is achieved by summing the results of simulating n Bernoulli-distributed random variables.

Scalar Output

pystats.rbinom(n_trials: int, prob: float) → float

Random sampling function for the Binomial distribution.

Example

>>> pystats.rbinom(4, 0.4)
2
Parameters

  • n_trials (int) – The number of trials, a non-negative integral-valued input.

  • prob (float) – The probability parameter, a real-valued input.

Returns

A pseudo-random draw from the Binomial distribution.

List Output

pystats.rbinom(n: int, n_trials: int, prob: float) → List[float]

Random sampling function for the Binomial distribution.

Example

>>> pystats.rbinom(10, 4, 0.4)
[1, 4, 0, 2, 3, 2, 2, 2, 2, 1]
Parameters

  • n (int) – The number of output values.

  • n_trials (int) – The number of trials, a non-negative integral-valued input.

  • prob (float) – The probability parameter, a real-valued input.

Returns

A list of pseudo-random draws from the Binomial distribution.