.. _examples-discrete:

Discrete Distribution Examples
================================

Worked examples for common discrete distributions. For each example the
range, PMF, and expected moment values are given. All results were verified
against known analytical formulae.

.. contents:: On this page
   :local:
   :depth: 2

----

Bernoulli Distribution
-----------------------

The Bernoulli distribution models a single binary trial with success
probability :math:`p`.

**Parameters:** :math:`p = 0.6`

**Support:** 0, 1

.. code-block:: text

   Range:  0,1
   PMF:    0.6 if x == 1 else 0.4

**Known moments about origin:**

.. list-table::
   :header-rows: 1
   :widths: 15 85

   * - Order
     - Value
   * - mu_1
     - 0.6  (= p)
   * - mu_2
     - 0.6  (= p, since X^2 = X for Bernoulli)
   * - mu_3
     - 0.6
   * - mu_4
     - 0.6

**Known central moments:**

.. list-table::
   :header-rows: 1
   :widths: 15 85

   * - Order
     - Value
   * - mu_1
     - 0  (always)
   * - mu_2
     - 0.24  (= p*(1-p))

----

Binomial Distribution
----------------------

The Binomial distribution models the number of successes in :math:`n`
independent Bernoulli trials.

**Parameters:** :math:`n = 10`, :math:`p = 0.4`

**Support:** 0, 1, 2, …, 10

.. code-block:: text

   Range:  0,1,2,3,4,5,6,7,8,9,10
   PMF:    (factorial(10) / (factorial(x) * factorial(10-x))) * 0.4**x * 0.6**(10-x)

**Known analytical values:**

.. code-block:: text

   Mean      = n*p        = 4.0
   Variance  = n*p*(1-p)  = 2.4
   Std Dev   = sqrt(2.4)  ≈ 1.5492
   Skewness  = (1-2p)/sqrt(n*p*(1-p)) ≈ 0.1291
   Kurtosis  = 3 + (1-6p(1-p))/(n*p*(1-p)) ≈ 3.0583

----

Geometric Distribution (failures before first success)
-------------------------------------------------------

The Geometric distribution models the number of failures before the
first success, with support starting at 0.

**Parameters:** :math:`p = 0.3`

**Support:** 0, 1, 2, 3, …

.. code-block:: text

   Range:  0,1,2,3,...
   PMF:    0.3 * (0.7 ** x) if x >= 0 else 0

**Known analytical values:**

.. code-block:: text

   Mean      = (1-p)/p     = 7/3  ≈ 2.3333
   Variance  = (1-p)/p²    = 70/9 ≈ 7.7778
   Skewness  = (2-p)/sqrt(1-p) ≈ 2.031
   Kurtosis  = 9 + p²/(1-p)   ≈ 9.129

----

Geometric Distribution (trials until first success)
----------------------------------------------------

Alternative parameterisation: support starts at 1.

**Parameters:** :math:`p = 0.3`

.. code-block:: text

   Range:  1,2,3,4,...
   PMF:    0.3 * (0.7 ** (x-1)) if x >= 1 else 0

**Known analytical values:**

.. code-block:: text

   Mean      = 1/p         = 10/3 ≈ 3.3333
   Variance  = (1-p)/p²    ≈ 7.7778  (same as above)

----

Poisson Distribution
---------------------

The Poisson distribution models the number of events in a fixed interval
when events occur at constant rate :math:`\lambda`.

**Parameters:** :math:`\lambda = 3`

**Support:** 0, 1, 2, 3, …

.. code-block:: text

   Range:  0,1,2,3,...
   PMF:    (exp(-3) * 3**x) / factorial(x) if x >= 0 else 0

**Known analytical values:**

.. code-block:: text

   Mean      = λ           = 3.0
   Variance  = λ           = 3.0
   Skewness  = 1/sqrt(λ)   ≈ 0.5774
   Kurtosis  = 3 + 1/λ     = 3.3333

.. tip::

   The Poisson distribution's PMF decays very quickly for large :math:`x`
   relative to :math:`\lambda`. With :math:`\lambda = 3`, 200 terms
   captures more than :math:`10^{-200}` of the total probability.

----

Negative Binomial Distribution
--------------------------------

The Negative Binomial models the number of failures before the
:math:`r`-th success.

**Parameters:** :math:`r = 2`, :math:`p = 0.5`

**Support:** 0, 1, 2, 3, …

.. code-block:: text

   Range:  0,1,2,3,...
   PMF:    (factorial(x+1) / (factorial(x) * factorial(1))) * 0.5**2 * 0.5**x if x >= 0 else 0

**Known analytical values:**

.. code-block:: text

   Mean      = r*(1-p)/p   = 2
   Variance  = r*(1-p)/p²  = 4

----

Uniform Discrete Distribution
------------------------------

Equal probability over a finite set of values.

**Parameters:** values 1 through 6 (fair die)

.. code-block:: text

   Range:  1,2,3,4,5,6
   PMF:    1/6

**Known analytical values:**

.. code-block:: text

   Mean      = (n+1)/2     = 3.5
   Variance  = (n²-1)/12   = 35/12 ≈ 2.9167
   Skewness  = 0           (symmetric)
   Kurtosis  = 3 - (6(n²+1))/(5(n²-1)) ≈ 2.5143

----

Tips for All Discrete Distributions
-------------------------------------

* Set the moment reference to **Mean (a = μ)** and order ≥ 4 to see
  skewness and kurtosis in the Statistics tab.
* Cross-check the first moment (μ₁) against the known analytical mean
  before trusting higher-order results.
* For the Poisson distribution with small :math:`\lambda`, increasing
  **Calc precision** to 10 helps accuracy for the 4th moment and above.
* If validation fails with sum ≠ 1, verify the PMF guard (``if x >= 0
  else 0``) and check for typos in the exponent.

See also
--------

* :doc:`/user-guide/discrete-rv` — full DRV user guide
* :doc:`/theory/convergence` — how infinite series are handled
* :doc:`advanced-examples` — mixture distributions and custom PMFs