.. _examples-continuous:

Continuous Distribution Examples
==================================

Worked examples for standard continuous distributions. Expected values
are the known analytical results; ToFUL results match to the displayed
precision.

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

----

Uniform Distribution
---------------------

**Parameters:** :math:`a = 0`, :math:`b = 1`

.. code-block:: text

   Lower:  0
   Upper:  1
   PDF:    1 if 0 <= x <= 1 else 0

**Known analytical values:**

.. code-block:: text

   Mean      = (a+b)/2         = 0.5
   Variance  = (b-a)²/12       = 1/12 ≈ 0.0833
   Skewness  = 0               (symmetric)
   Excess kurtosis = −6/5      = −1.2  (platykurtic)

**Raw moments about origin:**

.. math::

   \mu'_r = \frac{1}{r+1}

So :math:`\mu'_1 = 1/2`, :math:`\mu'_2 = 1/3`, :math:`\mu'_3 = 1/4`,
:math:`\mu'_4 = 1/5`.

----

Exponential Distribution
-------------------------

**Parameters:** :math:`\lambda = 2`

.. code-block:: text

   Lower:  0
   Upper:  inf
   PDF:    2 * exp(-2*x) if x >= 0 else 0

**Known analytical values:**

.. code-block:: text

   Mean      = 1/λ         = 0.5
   Variance  = 1/λ²        = 0.25
   Std Dev   = 0.5
   Skewness  = 2           (strong right skew)
   Excess kurtosis = 6     (heavy tails)

**Raw moments about origin:**

.. math::

   \mu'_r = \frac{r!}{\lambda^r}

So :math:`\mu'_1 = 0.5`, :math:`\mu'_2 = 0.5`, :math:`\mu'_3 = 0.75`,
:math:`\mu'_4 = 1.5`.

.. note::

   For this distribution, ToFUL uses Gauss-Laguerre quadrature
   (domain :math:`[0, \infty)`) and typically achieves near-exact results.
   The method shown will be ``gauss-laguerre+quad``.

----

Standard Normal Distribution
-----------------------------

**Parameters:** :math:`\mu = 0`, :math:`\sigma = 1`

.. code-block:: text

   Lower:  -inf
   Upper:  inf
   PDF:    exp(-(x**2)/2) / sqrt(2*pi)

**Known analytical values:**

.. code-block:: text

   Mean      = 0
   Variance  = 1
   Skewness  = 0   (symmetric)
   Kurtosis  = 3   (excess = 0, mesokurtic)

**Raw moments about origin:**

All odd raw moments are 0. Even raw moments satisfy:

.. math::

   \mu'_{2k} = (2k-1)!! = 1 \cdot 3 \cdot 5 \cdots (2k-1)

So :math:`\mu'_2 = 1`, :math:`\mu'_4 = 3`, :math:`\mu'_6 = 15`.

.. tip::

   The Normal PDF has no ``if/else`` guard, so ToFUL's SymPy path
   computes these moments exactly. The method shown will be ``sympy-exact``
   and all values will be precise integers or exact fractions.

----

Normal Distribution (general)
-------------------------------

**Parameters:** :math:`\mu = 3`, :math:`\sigma = 2`

.. code-block:: text

   Lower:  -inf
   Upper:  inf
   PDF:    exp(-((x-3)**2) / 8) / (2 * sqrt(2*pi))

**Known analytical values:**

.. code-block:: text

   Mean      = 3
   Variance  = 4
   Std Dev   = 2
   Skewness  = 0
   Excess kurtosis = 0

----

Gamma Distribution
-------------------

**Parameters:** :math:`\alpha = 3` (shape), :math:`\beta = 1` (rate)

.. code-block:: text

   Lower:  0
   Upper:  inf
   PDF:    (x**2 * exp(-x)) / 2  if x >= 0 else 0

(This is the Gamma(3,1) PDF normalised by :math:`\Gamma(3) = 2! = 2`.)

**Known analytical values:**

.. code-block:: text

   Mean      = α/β         = 3
   Variance  = α/β²        = 3
   Skewness  = 2/sqrt(α)   ≈ 1.1547
   Excess kurtosis = 6/α   = 2

**Raw moments about origin:**

.. math::

   \mu'_r = \frac{\Gamma(\alpha + r)}{\beta^r\,\Gamma(\alpha)}
   = \frac{(\alpha+r-1)!}{(\alpha-1)!\,\beta^r}

For :math:`\alpha = 3`, :math:`\beta = 1`:
:math:`\mu'_1 = 3`, :math:`\mu'_2 = 12`, :math:`\mu'_3 = 60`,
:math:`\mu'_4 = 360`.

----

Beta Distribution
------------------

**Parameters:** :math:`\alpha = 2`, :math:`\beta = 3`

The normalising constant :math:`B(\alpha,\beta)^{-1} = \Gamma(\alpha+\beta)/(\Gamma(\alpha)\Gamma(\beta)) = 12`.

.. code-block:: text

   Lower:  0
   Upper:  1
   PDF:    12 * x * (1-x)**2  if  0 <= x <= 1  else  0

**Known analytical values:**

.. code-block:: text

   Mean      = α/(α+β)           = 0.4
   Variance  = αβ/((α+β)²(α+β+1)) = 0.04
   Skewness  ≈ 0.5963
   Excess kurtosis ≈ −0.1429

----

Triangular Distribution
------------------------

**Parameters:** lower = 0, upper = 2, mode = 1

.. code-block:: text

   Lower:  0
   Upper:  2
   PDF:    x if 0<=x<=1 else (2-x) if 1<x<=2 else 0

**Known analytical values:**

.. code-block:: text

   Mean      = (a + b + c)/3     = 1
   Variance  = (a²+b²+c²-ab-ac-bc)/18 = 1/6 ≈ 0.1667
   Skewness  = 0  (symmetric for this parameterisation)

----

Cauchy Distribution
--------------------

The Cauchy distribution is the classic example of a distribution with
**no finite moments of any order**.

.. code-block:: text

   Lower:  -inf
   Upper:  inf
   PDF:    1 / (pi * (1 + x**2))

.. warning::

   ToFUL will validate that the PDF integrates to 1 (it does), but all
   moment computations will fail to converge. The Convergence tab will
   show ``partial-sum`` with an uncertain flag for all orders. This is
   the correct mathematical result — the integrals are genuinely infinite.

----

Laplace Distribution
---------------------

**Parameters:** :math:`\mu = 0`, :math:`b = 1`

.. code-block:: text

   Lower:  -inf
   Upper:  inf
   PDF:    exp(-abs(x)) / 2

**Known analytical values:**

.. code-block:: text

   Mean      = 0
   Variance  = 2b²         = 2
   Skewness  = 0           (symmetric)
   Excess kurtosis = 3     (heavier tails than Normal)

See also
--------

* :doc:`/user-guide/continuous-rv` — CRV user guide with quadrature details
* :doc:`/theory/skewness-kurtosis` — interpreting shape measures
* :doc:`advanced-examples` — mixture models and truncated distributions