maths.monte_carlo
=================

.. py:module:: maths.monte_carlo

.. autoapi-nested-parse::

   @author: MatteoRaso



Functions
---------

.. autoapisummary::

   maths.monte_carlo.area_under_curve_estimator
   maths.monte_carlo.area_under_line_estimator_check
   maths.monte_carlo.pi_estimator
   maths.monte_carlo.pi_estimator_using_area_under_curve


Module Contents
---------------

.. py:function:: area_under_curve_estimator(iterations: int, function_to_integrate: collections.abc.Callable[[float], float], min_value: float = 0.0, max_value: float = 1.0) -> float

   An implementation of the Monte Carlo method to find area under
      a single variable non-negative real-valued continuous function,
      say f(x), where x lies within a continuous bounded interval,
      say [min_value, max_value], where min_value and max_value are
      finite numbers
   1. Let x be a uniformly distributed random variable between min_value to
      max_value
   2. Expected value of f(x) =
      (integrate f(x) from min_value to max_value)/(max_value - min_value)
   3. Finding expected value of f(x):
       a. Repeatedly draw x from uniform distribution
       b. Evaluate f(x) at each of the drawn x values
       c. Expected value = average of the function evaluations
   4. Estimated value of integral = Expected value * (max_value - min_value)
   5. Returns estimated value


.. py:function:: area_under_line_estimator_check(iterations: int, min_value: float = 0.0, max_value: float = 1.0) -> None

   Checks estimation error for area_under_curve_estimator function
   for f(x) = x where x lies within min_value to max_value
   1. Calls "area_under_curve_estimator" function
   2. Compares with the expected value
   3. Prints estimated, expected and error value


.. py:function:: pi_estimator(iterations: int)

   An implementation of the Monte Carlo method used to find pi.
   1. Draw a 2x2 square centred at (0,0).
   2. Inscribe a circle within the square.
   3. For each iteration, place a dot anywhere in the square.
      a. Record the number of dots within the circle.
   4. After all the dots are placed, divide the dots in the circle by the total.
   5. Multiply this value by 4 to get your estimate of pi.
   6. Print the estimated and numpy value of pi


.. py:function:: pi_estimator_using_area_under_curve(iterations: int) -> None

   Area under curve y = sqrt(4 - x^2) where x lies in 0 to 2 is equal to pi