add example to estimate area under line using montecarlo (TheAlgorithms#1782)

* add example to estimate area under line using montecarlo

* separate estimate func and print statements

* use mean from stats package

* avoid creating extra variable

* min_value: float=0.0, max_value: float=1.0

* Update montecarlo.py

* Update montecarlo.py

* Rename montecarlo.py to monte_carlo.py

* Update monte_carlo.py

Co-authored-by: Christian Clauss <cclauss@me.com>

Author: singlav

maths/monte_carlo.py

@@ -0,0 +1,74 @@
+"""
+@author: MatteoRaso
+"""
+from numpy import pi, sqrt
+from random import uniform
+from statistics import mean
+
+
+def 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
+    """
+    # A local function to see if a dot lands in the circle.
+    def in_circle(x: float, y: float) -> bool:
+        distance_from_centre = sqrt((x ** 2) + (y ** 2))
+        # Our circle has a radius of 1, so a distance
+        # greater than 1 would land outside the circle.
+        return distance_from_centre <= 1
+
+    # The proportion of guesses that landed in the circle
+    proportion = mean(
+        int(in_circle(uniform(-1.0, 1.0), uniform(-1.0, 1.0))) for _ in range(iterations)
+    )
+    # The ratio of the area for circle to square is pi/4.
+    pi_estimate = proportion * 4
+    print("The estimated value of pi is ", pi_estimate)
+    print("The numpy value of pi is ", pi)
+    print("The total error is ", abs(pi - pi_estimate))
+
+
+def area_under_line_estimator(iterations: int,
+                              min_value: float=0.0,
+                              max_value: float=1.0) -> float:
+    """
+    An implementation of the Monte Carlo method to find area under
+       y = x where x lies between min_value to max_value
+    1. Let x be a uniformly distributed random variable between min_value to max_value
+    2. Expected value of x = integration of x from min_value to max_value
+    3. Finding expected value of x:
+        a. Repeatedly draw x from uniform distribution
+        b. Expected value = average of those values
+    4. Actual value = 1/2
+    5. Returns estimated value
+    """
+    return mean(uniform(min_value, max_value) for _ in range(iterations))
+
+
+def area_under_line_estimator_check(iterations: int) -> None:
+    """
+    Checks estimation error for area_under_line_estimator func
+    1. Calls "area_under_line_estimator" function
+    2. Compares with the expected value
+    3. Prints estimated, expected and error value
+    """
+    estimate = area_under_line_estimator(iterations)
+    print("******************")
+    print("Estimating area under y=x where x varies from 0 to 1")
+    print("Expected value is ", 0.5)
+    print("Estimated value is ", estimate)
+    print("Total error is ", abs(estimate - 0.5))
+    print("******************")
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()