feat: added poisson_dist.cpp (#739)

* added poisson_dist.cpp

* https://en.wikipedia.org/wiki/Poisson_distribution

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

probability/poisson_dist.cpp

@@ -0,0 +1,66 @@
+#include <iostream>
+#include <cmath>
+
+// The Poisson distribution counts how many
+// events occur over a set time interval
+// https://en.wikipedia.org/wiki/Poisson_distribution
+
+// calculate the events per unit time
+// e.g 5 dollars every 2 mins = 5 / 2 = 2.5
+
+double poisson_rate(double events, double timeframe) {
+    return events / timeframe;
+}
+
+// calculate the expected value over a time
+// e.g rate of 2.5 over 10 mins = 2.5 x 10 = 25
+
+double poisson_expected(double rate, double time) {
+    return rate * time;
+}
+
+// find the factorial of a given number
+
+double fact(double x) {
+    double x_fact = x;
+    for (int i = x - 1; i > 0; i--) {
+        x_fact *= i;
+    }
+
+    if (x_fact <= 0) {
+        x_fact = 1;
+    }
+    return x_fact;
+}
+
+// find the probability of x successes in a Poisson dist
+
+double poisson_x_successes(double expected, double x) {
+    return (pow(expected, x) * exp(-expected)) / fact(x);
+}
+
+// probability of a success in range for Poisson dist (inclusive, inclusive)
+
+double poisson_range_successes(double expected, double lower, double upper) {
+    double probability = 0;
+    for (int i = lower; i <= upper; i++) {
+        probability += poisson_x_successes(expected, i);
+    }
+    return probability;
+}
+
+int main() {
+    double rate, expected;
+    rate = poisson_rate(3, 1);
+    std::cout << "Poisson rate : " << rate << std::endl;
+
+    expected = poisson_expected(rate, 2);
+    std::cout << "Poisson expected : " << expected << std::endl;
+
+    std::cout << "Poisson 0 successes : "
+        <<poisson_x_successes(expected, 0) <<std::endl;
+    std::cout << "Poisson 0-8 successes : "
+        << poisson_range_successes(expected, 0, 8) <<std::endl;
+
+    return 0;
+}