binomial_dist.cpp File Reference

Binomial distribution example More...

#include <cmath>
#include <iostream>

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Functions
double	binomial_expected (double n, double p)
double	binomial_variance (double n, double p)
double	binomial_standard_deviation (double n, double p)
double	nCr (double n, double r)
double	binomial_x_successes (double n, double p, double x)
double	binomial_range_successes (double n, double p, double lower_bound, double upper_bound)
int	main ()

Detailed Description

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

Function Documentation

binomial_expected()

double binomial_expected	(	double	n,
		double	p
	)

finds the expected value of a binomial distribution

Parameters

[in]	n
[in]	p

Returns

\(\mu=np\)

{ return n * p; }

binomial_range_successes()

double binomial_range_successes	(	double	n,
		double	p,
		double	lower_bound,
		double	upper_bound
	)

calculates the probability of a result within a range (inclusive, inclusive)

Returns

\(\displaystyle \left.P(n,p)\right|_{x_0}^{x_1} = \sum_{i=x_0}^{x_1} P(i) =\sum_{i=x_0}^{x_1} {n\choose i} p^i (1-p)^{n-i}\)

                                                    {
    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;
}

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binomial_standard_deviation()

double binomial_standard_deviation	(	double	n,
		double	p
	)

finds the standard deviation of the binomial distribution

Parameters

[in]	n
[in]	p

Returns

\(\sigma = \sqrt{\sigma^2} = \sqrt{n\cdot p\cdot (1-p)}\)

                                                       {
    return std::sqrt(binomial_variance(n, p));
}

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binomial_variance()

double binomial_variance	(	double	n,
		double	p
	)

finds the variance of the binomial distribution

Parameters

[in]	n
[in]	p

Returns

\(\sigma^2 = n\cdot p\cdot (1-p)\)

{ return n * p * (1 - p); }

binomial_x_successes()

double binomial_x_successes	(	double	n,
		double	p,
		double	x
	)

calculates the probability of exactly x successes

Returns

\(\displaystyle P(n,p,x) = {n\choose x} p^x (1-p)^{n-x}\)

                                                          {
    return nCr(n, x) * std::pow(p, x) * std::pow(1 - p, n - x);
}

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main()

int main

(

void

)

main function

           {
    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;
}

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nCr()

double nCr	(	double	n,
		double	r
	)

Computes n choose r

Parameters

[in]	n
[in]	r

Returns

\(\displaystyle {n\choose r} = \frac{n!}{r!(n-r)!} = \frac{n\times(n-1)\times(n-2)\times\cdots(n-r)}{r!} \)

                               {
    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;
}