.. _theory-moments:

Statistical Moments
====================

Statistical moments are a family of summary statistics that describe the
shape of a probability distribution. ToFUL computes the *r*-th moment of
a random variable X about a reference point *a*, defined as:

.. math::

   \mu_r(a) = E\left[(X - a)^r\right]

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

----

Definition
----------

For a **discrete random variable** with PMF :math:`P(X = x)`:

.. math::

   \mu_r(a) = \sum_{x} (x - a)^r \cdot P(X = x)

For a **continuous random variable** with PDF :math:`f(x)`:

.. math::

   \mu_r(a) = \int_{-\infty}^{\infty} (x - a)^r \cdot f(x)\, dx

The choice of reference point *a* determines the *type* of moment.

----

Types of Moments
-----------------

Raw Moments (a = 0)
~~~~~~~~~~~~~~~~~~~~

Setting :math:`a = 0` gives the *raw moments* (also called moments
about the origin):

.. math::

   \mu'_r = E[X^r]

The first few raw moments have direct interpretations:

+-------+--------------------+-------------------------------------------+
| Order | Raw moment         | Interpretation                            |
+=======+====================+===========================================+
| r = 1 | :math:`\mu'_1`     | **Mean** mu = E[X]                        |
+-------+--------------------+-------------------------------------------+
| r = 2 | :math:`\mu'_2`     | E[X^2] -- used to compute variance        |
+-------+--------------------+-------------------------------------------+
| r = 3 | :math:`\mu'_3`     | E[X^3]                                    |
+-------+--------------------+-------------------------------------------+
| r = 4 | :math:`\mu'_4`     | E[X^4]                                    |
+-------+--------------------+-------------------------------------------+

Central Moments (a = μ)
~~~~~~~~~~~~~~~~~~~~~~~~

Setting :math:`a = \mu` gives the *central moments*:

.. math::

   \mu_r = E\left[(X - \mu)^r\right]

Central moments measure deviations from the mean and are therefore
invariant to shifts in the distribution. Key central moments:

+-------+-----------------------+-----------------------------------------+
| Order | Central moment        | Interpretation                          |
+=======+=======================+=========================================+
| r = 1 | :math:`\mu_1 = 0`     | Always zero by definition of mean       |
+-------+-----------------------+-----------------------------------------+
| r = 2 | :math:`\mu_2`         | **Variance** sigma^2                    |
+-------+-----------------------+-----------------------------------------+
| r = 3 | :math:`\mu_3`         | Raw material for skewness               |
+-------+-----------------------+-----------------------------------------+
| r = 4 | :math:`\mu_4`         | Raw material for kurtosis               |
+-------+-----------------------+-----------------------------------------+

Moments about a Custom Point (a = c)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For any constant :math:`c`, the moment about :math:`c` interpolates
between raw moments (c = 0) and central moments (c = μ). This is useful
when a distribution's natural reference point differs from both.

----

Relationships Between Moment Types
------------------------------------

Central moments can be expressed in terms of raw moments using the
binomial theorem:

.. math::

   \mu_r = \sum_{k=0}^{r} \binom{r}{k} (-\mu)^{r-k} \mu'_k

The first few expansions:

.. math::

   \mu_2 = \mu'_2 - (\mu'_1)^2 = E[X^2] - (E[X])^2

.. math::

   \mu_3 = \mu'_3 - 3\mu'_2\mu'_1 + 2(\mu'_1)^3

.. math::

   \mu_4 = \mu'_4 - 4\mu'_3\mu'_1 + 6\mu'_2(\mu'_1)^2 - 3(\mu'_1)^4

ToFUL does not use these formulae internally — it computes each moment
type directly from the definition, which avoids catastrophic cancellation
for large or broad distributions.

----

Existence of Moments
---------------------

Not all distributions have finite moments of all orders. A moment of
order :math:`r` exists (is finite) if and only if :math:`E[|X|^r] < \infty`.

For distributions with heavy tails, high-order moments may be infinite:

* **Cauchy distribution** — no finite moments of any order.
* **Pareto distribution** — moments of order :math:`r \geq \alpha` are
  infinite, where :math:`\alpha` is the tail index.
* **Student-t with ν degrees of freedom** — moments of order :math:`r \geq \nu`
  are infinite.

When a moment does not exist, the integration or sum will either return
a very large number or fail to converge. The Convergence tab will report
the method as ``partial-sum`` with a note about divergence.

----

Moment Generating Function
---------------------------

The *moment generating function* (MGF) of X is:

.. math::

   M_X(t) = E\left[e^{tX}\right]

When it exists in a neighbourhood of :math:`t = 0`, the raw moments are
obtained by differentiation:

.. math::

   \mu'_r = M_X^{(r)}(0) = \left.\frac{d^r}{dt^r} M_X(t)\right|_{t=0}

ToFUL does not use MGFs directly, but the SymPy symbolic path essentially
performs the equivalent symbolic integration.

----

Standardised Moments
---------------------

The *standardised r-th moment* divides the central moment by the
appropriate power of the standard deviation:

.. math::

   \tilde{\mu}_r = \frac{\mu_r}{\sigma^r}

The two most commonly standardised moments are skewness (:math:`\tilde{\mu}_3`)
and kurtosis (:math:`\tilde{\mu}_4`). See :doc:`skewness-kurtosis` for
a detailed treatment.

See also
--------

* :doc:`raw-vs-central` — choosing between raw and central moments
* :doc:`skewness-kurtosis` — shape characterisation via standardised moments
* :doc:`convergence` — how ToFUL evaluates the sums and integrals