Bridge Statistics
=================

.. module:: microscopic_gating.statistics

This module implements statistical models for bridge counts and gate probabilities.

Bridge Count Model
------------------

.. autoclass:: BridgeCountModel
   :members:
   :member-order: bysource
   :show-inheritance:

Gate Model
----------

.. autoclass:: GateModel
   :members:
   :member-order: bysource
   :show-inheritance:

Theory Background
-----------------

Bridge counts follow a Poisson distribution with intensity :math:`\lambda`:

.. math::

   P(n_b) = \frac{\lambda^{n_b} e^{-\lambda}}{n_b!}

where :math:`\lambda = N_{\text{pair}} \cdot P_{\text{bridge}}(\phi)`.

Gate Open Probability
~~~~~~~~~~~~~~~~~~~~~

The probability of zero bridges (gate open):

.. math::

   P_{\text{open}} = e^{-\lambda}

Gate Closed Probability
~~~~~~~~~~~~~~~~~~~~~~~

The probability of at least one bridge (gate closed):

.. math::

   P_{\text{closed}} = 1 - e^{-\lambda}