#### Binomial Dist

/**
* @file
* @brief [Binomial
* distribution](https://en.wikipedia.org/wiki/Binomial_distribution) example
*
* The binomial distribution models the number of
* successes in a sequence of n independent events
*
* Summary of variables used:
* * n : number of trials
* * p : probability of success
* * x : desired successes
*/
#include <cmath>
#include <iostream>

/** finds the expected value of a binomial distribution
* \param [in] n
* \param [in] p
* \returns \f$\mu=np\f$
*/
double binomial_expected(double n, double p) { return n * p; }

/** finds the variance of the binomial distribution
* \param [in] n
* \param [in] p
* \returns \f$\sigma^2 = n\cdot p\cdot (1-p)\f$
*/
double binomial_variance(double n, double p) { return n * p * (1 - p); }

/** finds the standard deviation of the binomial distribution
* \param [in] n
* \param [in] p
* \returns \f$\sigma = \sqrt{\sigma^2} = \sqrt{n\cdot p\cdot (1-p)}\f$
*/
double binomial_standard_deviation(double n, double p) {
return std::sqrt(binomial_variance(n, p));
}

/** Computes n choose r
* \param [in] n
* \param [in] r
* \returns \f$\displaystyle {n\choose r} = * \frac{n!}{r!(n-r)!} = \frac{n\times(n-1)\times(n-2)\times\cdots(n-r)}{r!} * \f$
*/
double nCr(double n, double r) {
double numerator = n;
double denominator = r;

for (int i = n - 1; i >= ((n - r) + 1); i--) {
numerator *= i;
}

for (int i = 1; i < r; i++) {
denominator *= i;
}

return numerator / denominator;
}

/** calculates the probability of exactly x successes
* \returns \f$\displaystyle P(n,p,x) = {n\choose x} p^x (1-p)^{n-x}\f$
*/
double binomial_x_successes(double n, double p, double x) {
return nCr(n, x) * std::pow(p, x) * std::pow(1 - p, n - x);
}

/** calculates the probability of a result within a range (inclusive, inclusive)
* \returns \f$\displaystyle \left.P(n,p)\right|_{x_0}^{x_1} = * \sum_{i=x_0}^{x_1} P(i) * =\sum_{i=x_0}^{x_1} {n\choose i} p^i (1-p)^{n-i}\f$
*/
double binomial_range_successes(double n, double p, double lower_bound,
double upper_bound) {
double probability = 0;
for (int i = lower_bound; i <= upper_bound; i++) {
probability += nCr(n, i) * std::pow(p, i) * std::pow(1 - p, n - i);
}
return probability;
}

/** main function */
int main() {
std::cout << "expected value : " << binomial_expected(100, 0.5)
<< std::endl;

std::cout << "variance : " << binomial_variance(100, 0.5) << std::endl;

std::cout << "standard deviation : "
<< binomial_standard_deviation(100, 0.5) << std::endl;

std::cout << "exactly 30 successes : " << binomial_x_successes(100, 0.5, 30)
<< std::endl;

std::cout << "45 or more successes : "
<< binomial_range_successes(100, 0.5, 45, 100) << std::endl;

return 0;
}