From e3566d1c21dc368762f9c4c7986db85966253514 Mon Sep 17 00:00:00 2001 From: Isaac Shoebottom Date: Mon, 22 Apr 2024 12:52:11 -0300 Subject: [PATCH] Update distribution.py --- .idea/.gitignore | 2 + .idea/casio-calculator.iml | 4 +- Pipfile | 1 - Pipfile.lock | 3 +- distribution.py | 744 +++++++++++++++++++++---------------- 5 files changed, 436 insertions(+), 318 deletions(-) diff --git a/.idea/.gitignore b/.idea/.gitignore index 13566b8..a9d7db9 100644 --- a/.idea/.gitignore +++ b/.idea/.gitignore @@ -6,3 +6,5 @@ # Datasource local storage ignored files /dataSources/ /dataSources.local.xml +# GitHub Copilot persisted chat sessions +/copilot/chatSessions diff --git a/.idea/casio-calculator.iml b/.idea/casio-calculator.iml index d9a405f..7c56d14 100644 --- a/.idea/casio-calculator.iml +++ b/.idea/casio-calculator.iml @@ -1,7 +1,9 @@ - + + + diff --git a/Pipfile b/Pipfile index 08fb404..0757494 100644 --- a/Pipfile +++ b/Pipfile @@ -9,4 +9,3 @@ name = "pypi" [requires] python_version = "3.11" -python_full_version = "3.11.7" diff --git a/Pipfile.lock b/Pipfile.lock index 1a475dd..54a7078 100644 --- a/Pipfile.lock +++ b/Pipfile.lock @@ -1,11 +1,10 @@ { "_meta": { "hash": { - "sha256": "bc82cd27f07d4e24b750064464bbc233a141778868b9a387125705e2d4e8a830" + "sha256": "ed6d5d614626ae28e274e453164affb26694755170ccab3aa5866f093d51d3e4" }, "pipfile-spec": 6, "requires": { - "python_full_version": "3.11.7", "python_version": "3.11" }, "sources": [ diff --git a/distribution.py b/distribution.py index 486c29f..6b18bde 100644 --- a/distribution.py +++ b/distribution.py @@ -1,420 +1,536 @@ import math +import statistics def factorial(n): - """ - Computes the factorial of a number. - :param n: The number to compute the factorial of. - :return: Returns the factorial of the number. - """ - if n == 0: - return 1 - for i in range(1, n): - n *= i - return n + """ + Computes the factorial of a number. + :param n: The number to compute the factorial of. + :return: Returns the factorial of the number. + """ + if n == 0: + return 1 + for i in range(1, n): + n *= i + return n def combination(n, r): - """ - Computes the combination of n choose r. - :param n: The number of items. - :param r: The number of items to choose. - :return: Returns the number of ways to choose r items from n items. - """ - return factorial(n) / (factorial(r) * factorial(n - r)) + """ + Computes the combination of n choose r. + :param n: The number of items. + :param r: The number of items to choose. + :return: Returns the number of ways to choose r items from n items. + """ + return factorial(n) / (factorial(r) * factorial(n - r)) def bnd(x, n, p): - """ - Computes the binomial distribution. - :param x: Number of successes. - :param n: Number of trials. - :param p: Probability of success. - :return: Returns the probability of getting x successes in n trials. - """ - return combination(n, x) * p ** x * (1 - p) ** (n - x) + """ + Computes the binomial distribution. + :param x: Number of successes. + :param n: Number of trials. + :param p: Probability of success. + :return: Returns the probability of getting x successes in n trials. + """ + return combination(n, x) * p ** x * (1 - p) ** (n - x) def bnd_mean(n, p): - """ - Computes the mean of the binomial distribution. - :param n: Number of trials. - :param p: Probability of success. - :return: Returns the mean of the binomial distribution. - """ - return n * p + """ + Computes the mean of the binomial distribution. + :param n: Number of trials. + :param p: Probability of success. + :return: Returns the mean of the binomial distribution. + """ + return n * p def bnd_var(n, p): - """ - Computes the variance of the binomial distribution. - :param n: Number of trials. - :param p: Probability of success. - :return: Returns the variance of the binomial distribution. - """ - return n * p * (1 - p) + """ + Computes the variance of the binomial distribution. + :param n: Number of trials. + :param p: Probability of success. + :return: Returns the variance of the binomial distribution. + """ + return n * p * (1 - p) def bnd_std(n, p): - """ - Computes the standard deviation of the binomial distribution. - :param n: Number of trials. - :param p: Probability of success. - :return: Returns the standard deviation of the binomial distribution. - """ - return bnd_var(n, p) ** 0.5 + """ + Computes the standard deviation of the binomial distribution. + :param n: Number of trials. + :param p: Probability of success. + :return: Returns the standard deviation of the binomial distribution. + """ + return bnd_var(n, p) ** 0.5 def bnd_leq(x, n, p): - """ - Computes the cumulative probability less than or equal to x successes in n trials. - :param x: Number of successes. - :param n: Number of trials. - :param p: Probability of success. - :return: Returns the cumulative probability less than or equal to x successes in n trials. - """ - return sum(bnd(i, n, p) for i in range(x + 1)) + """ + Computes the cumulative probability less than or equal to x successes in n trials. + :param x: Number of successes. + :param n: Number of trials. + :param p: Probability of success. + :return: Returns the cumulative probability less than or equal to x successes in n trials. + """ + return sum(bnd(i, n, p) for i in range(x + 1)) def bnd_lt(x, n, p): - """ - Computes the cumulative probability less than x successes in n trials. - :param x: Number of successes. - :param n: Number of trials. - :param p: Probability of success. - :return: Returns the cumulative probability less than x successes in n trials. - """ - return sum(bnd(i, n, p) for i in range(x)) + """ + Computes the cumulative probability less than x successes in n trials. + :param x: Number of successes. + :param n: Number of trials. + :param p: Probability of success. + :return: Returns the cumulative probability less than x successes in n trials. + """ + return sum(bnd(i, n, p) for i in range(x)) def bnd_geq(x, n, p): - """ - Computes the cumulative probability greater than or equal to x successes in n trials. - :param x: Number of successes. - :param n: Number of trials. - :param p: Probability of success. - :return: Returns the cumulative probability greater than or equal to x successes in n trials. - """ - return 1 - bnd_lt(x, n, p) + """ + Computes the cumulative probability greater than or equal to x successes in n trials. + :param x: Number of successes. + :param n: Number of trials. + :param p: Probability of success. + :return: Returns the cumulative probability greater than or equal to x successes in n trials. + """ + return 1 - bnd_lt(x, n, p) def bnd_gt(x, n, p): - """ - Computes the cumulative probability greater than x successes in n trials. - :param x: Number of successes. - :param n: Number of trials. - :param p: Probability of success. - :return: Returns the cumulative probability greater than x successes in n trials. - """ - return 1 - bnd_leq(x, n, p) + """ + Computes the cumulative probability greater than x successes in n trials. + :param x: Number of successes. + :param n: Number of trials. + :param p: Probability of success. + :return: Returns the cumulative probability greater than x successes in n trials. + """ + return 1 - bnd_leq(x, n, p) def gd(x, p, q=None): - """ - Computes the geometric distribution. - :param x: Number of trials until the first success. - :param p: Probability of success. - :param q: Probability of failure. - :return: Returns the probability of getting the first success on the xth trial. - """ - if q is None: - q = 1 - p - return q ** (x - 1) * p + """ + Computes the geometric distribution. + :param x: Number of trials until the first success. + :param p: Probability of success. + :param q: Probability of failure. + :return: Returns the probability of getting the first success on the xth trial. + """ + if q is None: + q = 1 - p + return q ** (x - 1) * p def gd_mean(p): - """ - Computes the mean of the geometric distribution. - :param p: Probability of success. - :return: Returns the mean of the geometric distribution. - """ - return 1 / p + """ + Computes the mean of the geometric distribution. + :param p: Probability of success. + :return: Returns the mean of the geometric distribution. + """ + return 1 / p def gd_var(p): - """ - Computes the variance of the geometric distribution. - :param p: Probability of success. - :return: Returns the variance of the geometric distribution. - """ - return (1 - p) / p ** 2 + """ + Computes the variance of the geometric distribution. + :param p: Probability of success. + :return: Returns the variance of the geometric distribution. + """ + return (1 - p) / p ** 2 def gd_std(p): - """ - Computes the standard deviation of the geometric distribution. - :param p: Probability of success. - :return: Returns the standard deviation of the geometric distribution. - """ - return gd_var(p) ** 0.5 + """ + Computes the standard deviation of the geometric distribution. + :param p: Probability of success. + :return: Returns the standard deviation of the geometric distribution. + """ + return gd_var(p) ** 0.5 def gd_leq(x, p, q=None): - """ - Computes the cumulative probability of getting upto x trials until the first success. - :param x: Number of trials until the first success. - :param p: Probability of success. - :param q: Probability of failure. - :return: Returns the cumulative probability of getting upto x trials until the first success. - """ - if q is not None: - return sum(gd(i, p, q) for i in range(1, x + 1)) - return sum(gd(i, p) for i in range(1, x + 1)) + """ + Computes the cumulative probability of getting upto x trials until the first success. + :param x: Number of trials until the first success. + :param p: Probability of success. + :param q: Probability of failure. + :return: Returns the cumulative probability of getting upto x trials until the first success. + """ + if q is not None: + return sum(gd(i, p, q) for i in range(1, x + 1)) + return sum(gd(i, p) for i in range(1, x + 1)) def gd_lt(x, p, q=None): - """ - Computes the cumulative probability of getting less than x trials until the first success. - :param x: Number of trials until the first success. - :param p: Probability of success. - :param q: Probability of failure. - :return: Returns the cumulative probability of getting less than x trials until the first success. - """ - if q is not None: - return sum(gd(i, p, q) for i in range(1, x)) - return sum(gd(i, p) for i in range(1, x)) + """ + Computes the cumulative probability of getting less than x trials until the first success. + :param x: Number of trials until the first success. + :param p: Probability of success. + :param q: Probability of failure. + :return: Returns the cumulative probability of getting less than x trials until the first success. + """ + if q is not None: + return sum(gd(i, p, q) for i in range(1, x)) + return sum(gd(i, p) for i in range(1, x)) def gd_geq(x, p, q=None): - """ - Computes the cumulative probability of getting from x trials until the first success. - :param x: Number of trials until the first success. - :param p: Probability of success. - :param q: Probability of failure. - :return: Returns the cumulative probability of getting from x trials until the first success. - """ - if q is not None: - return 1 - gd_lt(x, p, q) - return 1 - gd_leq(x, p) + """ + Computes the cumulative probability of getting from x trials until the first success. + :param x: Number of trials until the first success. + :param p: Probability of success. + :param q: Probability of failure. + :return: Returns the cumulative probability of getting from x trials until the first success. + """ + if q is not None: + return 1 - gd_lt(x, p, q) + return 1 - gd_leq(x, p) def gd_gt(x, p, q=None): - """ - Computes the cumulative probability of getting from x trials until the first success. - :param x: Number of trials until the first success. - :param p: Probability of success. - :param q: Probability of failure. - :return: Returns the cumulative probability of getting from x trials until the first success. - """ - if q is not None: - return 1 - gd_leq(x, p, q) - return 1 - gd_leq(x, p) + """ + Computes the cumulative probability of getting from x trials until the first success. + :param x: Number of trials until the first success. + :param p: Probability of success. + :param q: Probability of failure. + :return: Returns the cumulative probability of getting from x trials until the first success. + """ + if q is not None: + return 1 - gd_leq(x, p, q) + return 1 - gd_leq(x, p) def hgd(x, N, n, k): - """ - Computes the hyper geometric distribution. - :param x: Number of successes in the sample. - :param N: Number of items in the population. - :param n: Number of draws. - :param k: Number of successes in the population. - :return: Returns the probability of getting x successes in n draws from a population of size N with k successes. - """ - return (combination(k, x) * combination(N - k, n - x)) / combination(N, n) + """ + Computes the hyper geometric distribution. + :param x: Number of successes in the sample. + :param N: Number of items in the population. + :param n: Number of draws. + :param k: Number of successes in the population. + :return: Returns the probability of getting x successes in n draws from a population of size N with k successes. + """ + return (combination(k, x) * combination(N - k, n - x)) / combination(N, n) def hgd_mean(N, n, k): - """ - Computes the mean of the hyper geometric distribution. - :param N: Number of items in the population. - :param n: Number of draws. - :param k: Number of successes in the population. - :return: Returns the mean of the hyper geometric distribution. - """ - return n * (k / N) + """ + Computes the mean of the hyper geometric distribution. + :param N: Number of items in the population. + :param n: Number of draws. + :param k: Number of successes in the population. + :return: Returns the mean of the hyper geometric distribution. + """ + return n * (k / N) def hgd_var(N, n, k): - """ - Computes the variance of the hyper geometric distribution. - :param N: Number of items in the population. - :param n: Number of draws. - :param k: Number of successes in the population. - :return: Returns the variance of the hyper geometric distribution. - """ - return (n * k * (N - k) * (N - n)) / (N ** 2 * (N - 1)) + """ + Computes the variance of the hyper geometric distribution. + :param N: Number of items in the population. + :param n: Number of draws. + :param k: Number of successes in the population. + :return: Returns the variance of the hyper geometric distribution. + """ + return (n * k * (N - k) * (N - n)) / (N ** 2 * (N - 1)) def hgd_std(N, n, k): - """ - Computes the standard deviation of the hyper geometric distribution. - :param N: Number of items in the population. - :param n: Number of draws. - :param k: Number of successes in the population. - :return: Returns the standard deviation of the hyper geometric distribution. - """ - return hgd_var(N, n, k) ** 0.5 + """ + Computes the standard deviation of the hyper geometric distribution. + :param N: Number of items in the population. + :param n: Number of draws. + :param k: Number of successes in the population. + :return: Returns the standard deviation of the hyper geometric distribution. + """ + return hgd_var(N, n, k) ** 0.5 def hgd_leq(x, N, n, k): - """ - Computes the cumulative probability of getting upto x successes in n draws from a population of size N with k successes. - :param x: Number of successes in the sample. - :param N: Number of items in the population. - :param n: Number of draws. - :param k: Number of successes in the population. - :return: Returns the cumulative probability of getting upto x successes in n draws from a population of size N with k successes. - """ - return sum(hgd(i, N, n, k) for i in range(x + 1)) + """ + Computes the cumulative probability of getting upto x successes in n draws from a population of size N with k successes. + :param x: Number of successes in the sample. + :param N: Number of items in the population. + :param n: Number of draws. + :param k: Number of successes in the population. + :return: Returns the cumulative probability of getting upto x successes in n draws from a population of size N with k successes. + """ + return sum(hgd(i, N, n, k) for i in range(x + 1)) def hgd_lt(x, N, n, k): - """ - Computes the cumulative probability of getting less than x successes in n draws from a population of size N with k successes. - :param x: Number of successes in the sample. - :param N: Number of items in the population. - :param n: Number of draws. - :param k: Number of successes in the population. - :return: Returns the cumulative probability of getting less than x successes in n draws from a population of size N with k successes. - """ - return sum(hgd(i, N, n, k) for i in range(x)) + """ + Computes the cumulative probability of getting less than x successes in n draws from a population of size N with k successes. + :param x: Number of successes in the sample. + :param N: Number of items in the population. + :param n: Number of draws. + :param k: Number of successes in the population. + :return: Returns the cumulative probability of getting less than x successes in n draws from a population of size N with k successes. + """ + return sum(hgd(i, N, n, k) for i in range(x)) def hgd_geq(x, N, n, k): - """ - Computes the cumulative probability of getting from x successes in n draws from a population of size N with k successes. - :param x: Number of successes in the sample. - :param N: Number of items in the population. - :param n: Number of draws. - :param k: Number of successes in the population. - :return: Returns the cumulative probability of getting from x successes in n draws from a population of size N with k successes. - """ - return 1 - hgd_lt(x, N, n, k) + """ + Computes the cumulative probability of getting from x successes in n draws from a population of size N with k successes. + :param x: Number of successes in the sample. + :param N: Number of items in the population. + :param n: Number of draws. + :param k: Number of successes in the population. + :return: Returns the cumulative probability of getting from x successes in n draws from a population of size N with k successes. + """ + return 1 - hgd_lt(x, N, n, k) def hgd_gt(x, N, n, k): - """ - Computes the cumulative probability of getting from x successes in n draws from a population of size N with k successes. - :param x: Number of successes in the sample. - :param N: Number of items in the population. - :param n: Number of draws. - :param k: Number of successes in the population. - :return: Returns the cumulative probability of getting from x successes in n draws from a population of size N with k successes. - """ - return 1 - hgd_leq(x, N, n, k) + """ + Computes the cumulative probability of getting from x successes in n draws from a population of size N with k successes. + :param x: Number of successes in the sample. + :param N: Number of items in the population. + :param n: Number of draws. + :param k: Number of successes in the population. + :return: Returns the cumulative probability of getting from x successes in n draws from a population of size N with k successes. + """ + return 1 - hgd_leq(x, N, n, k) def pd(x, l): - """ - Computes the poisson distribution. - :param x: Number of occurrences. - :param l: Average number of occurrences. - :return: Returns the probability of getting x occurrences. - """ - return (l ** x * math.e ** -l) / factorial(x) + """ + Computes the poisson distribution. + :param x: Number of occurrences. + :param l: Average number of occurrences. + :return: Returns the probability of getting x occurrences. + """ + return (l ** x * math.e ** -l) / factorial(x) def pd_mean(l): - """ - Computes the mean of the poisson distribution. - :param l: Average number of occurrences. - :return: Returns the mean of the poisson distribution. - """ - return l + """ + Computes the mean of the poisson distribution. + :param l: Average number of occurrences. + :return: Returns the mean of the poisson distribution. + """ + return l def pd_var(l): - """ - Computes the variance of the poisson distribution. - :param l: Average number of occurrences. - :return: Returns the variance of the poisson distribution. - """ - return l + """ + Computes the variance of the poisson distribution. + :param l: Average number of occurrences. + :return: Returns the variance of the poisson distribution. + """ + return l def pd_std(l): - """ - Computes the standard deviation of the poisson distribution. - :param l: Average number of occurrences. - :return: Returns the standard deviation of the poisson distribution. - """ - return l ** 0.5 + """ + Computes the standard deviation of the poisson distribution. + :param l: Average number of occurrences. + :return: Returns the standard deviation of the poisson distribution. + """ + return l ** 0.5 def pd_leq(x, l): - """ - Computes the cumulative probability of getting upto x occurrences. - :param x: Number of occurrences. - :param l: Average number of occurrences. - :return: Returns the cumulative probability of getting upto x occurrences. - """ - return sum(pd(i, l) for i in range(x + 1)) + """ + Computes the cumulative probability of getting upto x occurrences. + :param x: Number of occurrences. + :param l: Average number of occurrences. + :return: Returns the cumulative probability of getting upto x occurrences. + """ + return sum(pd(i, l) for i in range(x + 1)) def pd_lt(x, l): - """ - Computes the cumulative probability of getting less than x occurrences. - :param x: Number of occurrences. - :param l: Average number of occurrences. - :return: Returns the cumulative probability of getting less than x occurrences. - """ - return sum(pd(i, l) for i in range(x)) + """ + Computes the cumulative probability of getting less than x occurrences. + :param x: Number of occurrences. + :param l: Average number of occurrences. + :return: Returns the cumulative probability of getting less than x occurrences. + """ + return sum(pd(i, l) for i in range(x)) def pd_geq(x, l): - """ - Computes the cumulative probability of getting from x occurrences. - :param x: Number of occurrences. - :param l: Average number of occurrences. - :return: Returns the cumulative probability of getting from x occurrences. - """ - return 1 - pd_lt(x, l) + """ + Computes the cumulative probability of getting from x occurrences. + :param x: Number of occurrences. + :param l: Average number of occurrences. + :return: Returns the cumulative probability of getting from x occurrences. + """ + return 1 - pd_lt(x, l) def pd_gt(x, l): - """ - Computes the cumulative probability of getting from x occurrences. - :param x: Number of occurrences. - :param l: Average number of occurrences. - :return: Returns the cumulative probability of getting from x occurrences. - """ - return 1 - pd_leq(x, l) + """ + Computes the cumulative probability of getting from x occurrences. + :param x: Number of occurrences. + :param l: Average number of occurrences. + :return: Returns the cumulative probability of getting from x occurrences. + """ + return 1 - pd_leq(x, l) + + +def sample_mean_e(u): + """ + Computes the expected value of the sample mean. + :param u: The population mean. + :return: Returns the expected value of the sample mean. + """ + return u + + +def sample_mean_std(u, n): + """ + Computes the standard deviation of the sample mean. + :param u: The population mean. + :param n: The sample size. + :return: Returns the standard deviation of the sample mean. + """ + return u / n ** 0.5 + + +def sample_mean_var(u, n): + """ + Computes the variance of the sample mean. + :param u: The population mean. + :param n: The sample size. + :return: Returns the variance of the sample mean. + """ + return (sample_mean_std(u, n) ** 2) / n + + +def z_score(x, u, s): + """ + Computes the z-score of a sample. + :param x: The sample mean. + :param u: The population mean. + :param s: The standard deviation of the sample mean. + :return: Returns the z-score of the sample. + """ + return (x - u) / s + + +def z_to_p(z): + """ + Computes the probability of a z-score. + :param z: The z-score. + :return: Returns the probability of the z-score. + """ + nd = statistics.NormalDist() + return nd.cdf(z) + + +def p_to_z(p): + """ + Computes the z-score of a probability. + :param p: The probability. + :return: Returns the z-score of the probability. + """ + nd = statistics.NormalDist() + return nd.inv_cdf(p) + + +def gamma(u, n): + """ + Computes the gamma of a sample. + :param u: The population mean. + :param n: The sample size. + :return: Returns the gamma of the sample. + """ + return sample_mean_var(u, n) / sample_mean_e(u) + + +def alpha(u, n): + """ + Computes the alpha of a sample. + :param u: The population mean. + :param n: The sample size. + :return: Returns the alpha of the sample. + """ + return sample_mean_e(u) / gamma(u, n) + + +def margin_of_error(a, s, n): + """ + Computes the margin of error of a sample. + :param a: The alpha of the sample. + :param s: The standard deviation of the sample mean. + :param n: The sample size. + :return: Returns the margin of error of the sample. + """ + return abs((p_to_z(a / 2)) * (s / (n ** 0.5))) + + +def confidence_interval(x, a, s, n): + """ + Computes the confidence interval of a sample. + :param x: The sample mean. + :param a: The alpha of the sample. + :param s: The standard deviation of the sample mean. + :param n: The sample size. + :return: Returns the confidence interval of the sample. + """ + return x - margin_of_error(a, s, n), x + margin_of_error(a, s, n) def man(): - """ - Prints the manual for the module. - """ - seperator = "-" * 20 - print("This module contains functions for computing the total probability of events.") - print("The functions are:") - print(seperator) - print("Binomial Distribution") - print("bnd(x, n, p)") - print("bnd_mean(n, p)") - print("bnd_var(n, p)") - print("bnd_std(n, p)") - print("bnd_leq(x, n, p)") - print("bnd_lt(x, n, p)") - print("bnd_geq(x, n, p)") - print("bnd_gt(x, n, p)") - print(seperator) - print("Geometric Distribution") - print("gd(x, p, q)") - print("gd_mean(p)") - print("gd_var(p)") - print("gd_std(p)") - print("gd_leq(x, p, q)") - print("gd_lt(x, p, q)") - print("gd_geq(x, p, q)") - print("gd_gt(x, p, q)") - print(seperator) - print("Hyper Geometric Distribution") - print("hgd(x, N, n, k)") - print("hgd_mean(N, n, k)") - print("hgd_var(N, n, k)") - print("hgd_std(N, n, k)") - print("hgd_leq(x, N, n, k)") - print("hgd_lt(x, N, n, k)") - print("hgd_geq(x, N, n, k)") - print("hgd_gt(x, N, n, k)") - print(seperator) - print("Poisson Distribution") - print("pd(x, l)") - print("pd_mean(l)") - print("pd_var(l)") - print("pd_std(l)") - print("pd_leq(x, l)") - print("pd_lt(x, l)") - print("pd_geq(x, l)") - print("pd_gt(x, l)") + """ + Prints the manual for the module. + """ + separator = "-" * 20 + print("This module contains functions for computing the total probability of events.") + print("The functions are:") + print(separator) + print("Binomial Distribution") + print("bnd(x, n, p)") + print("bnd_mean(n, p)") + print("bnd_var(n, p)") + print("bnd_std(n, p)") + print("bnd_leq(x, n, p)") + print("bnd_lt(x, n, p)") + print("bnd_geq(x, n, p)") + print("bnd_gt(x, n, p)") + print(separator) + print("Geometric Distribution") + print("gd(x, p, q)") + print("gd_mean(p)") + print("gd_var(p)") + print("gd_std(p)") + print("gd_leq(x, p, q)") + print("gd_lt(x, p, q)") + print("gd_geq(x, p, q)") + print("gd_gt(x, p, q)") + print(separator) + print("Hyper Geometric Distribution") + print("hgd(x, N, n, k)") + print("hgd_mean(N, n, k)") + print("hgd_var(N, n, k)") + print("hgd_std(N, n, k)") + print("hgd_leq(x, N, n, k)") + print("hgd_lt(x, N, n, k)") + print("hgd_geq(x, N, n, k)") + print("hgd_gt(x, N, n, k)") + print(separator) + print("Poisson Distribution") + print("pd(x, l)") + print("pd_mean(l)") + print("pd_var(l)") + print("pd_std(l)") + print("pd_leq(x, l)") + print("pd_lt(x, l)") + print("pd_geq(x, l)") + print("pd_gt(x, l)") + print(separator) + print("Sample Mean") + print("sample_mean_e(u)") + print("sample_mean_std(u, n)") + print("sample_mean_var(u, n)") + print("z_score(x, u, s)") + print("z_to_p(z)") + print("p_to_z(p)") + print("gamma(u, n)") + print("alpha(u, n)") + print("margin_of_error(a, s, n)") + print("confidence_interval(x, a, s, n)")