Bernoulli Distribution

Table of contents

Density Function

The density function of the Bernoulli distribution:

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

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

Scalar Input

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

Density function of the Bernoulli distribution.

Example

>>> pystats.dbern(1, 0.6)
0.6
Parameters

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

  • 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.dbern(x: List[int], prob: float, log: bool = False) → float

Density function of the Bernoulli distribution.

Example

>>> pystats.dbern([0, 1], 0.6)
[0.4, 0.6]
Parameters

  • x (List[int]) – A standard list 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 Bernoulli distribution:

\[\begin{split}F(x; p) = \sum_{z \leq x} f(z; p) = \begin{cases} 0 & \text{ if } x < 0 \\ 1-p & \text{ if } 0 \leq x < 1 \\ 1 & \text{ if } x \geq 1 \end{cases}\end{split}\]

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

Scalar Input

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

Distribution function of the Bernoulli distribution.

Example

>>> pystats.pbern(1, 0.6)
1.0
Parameters

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

  • 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.pbern(p: List[int], prob: float, log: bool = False) → float

Distribution function of the Bernoulli distribution.

Example

>>> pystats.pbern([0, 1], 0.6)
[0.4, 1.0]
Parameters

  • p (List[int]) – A standard list 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 Bernoulli distribution:

\[\begin{split}q(r; p) = \begin{cases} 0 & \text{ if } r \leq 1 - p \\ 1 & \text{ else } \end{cases}\end{split}\]

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

Scalar Input

pystats.qbern(q: float, prob: float) → float

Quantile function of the Bernoulli distribution.

Example

>>> pystats.qbern(0.5, 0.4)
0.0
Parameters

  • q (float) – A real-valued input.

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

Returns

The quantile function evaluated at q.

List Input

pystats.qbern(q: List[float], prob: float) → float

Quantile function of the Bernoulli distribution.

Example

>>> pystats.pbern([0.3, 0.7], 0.6)
[0.0, 1.0]
Parameters

  • q (List[float]) – A standard list 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 Bernoulli distribution is achieved via the inverse probability integral transform.

Scalar Output

pystats.rbern(prob: float) → float

Random sampling function for the Bernoulli distribution.

Example

>>> pystats.rbern(0.4)
0.0
Parameters

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

Returns

A pseudo-random draw from the Bernoulli distribution.

List Output

pystats.rbern(n: int, prob: float) → float

Random sampling function for the Bernoulli distribution.

Example

>>> pystats.rbern(10, 0.4)
[1.0, 0.0, 1.0, 0.0, 0.0, 1.0, 1.0, 1.0, 0.0, 0.0]
Parameters

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

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

Returns

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