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Update distribution.py

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Isaac Shoebottom 2024-04-22 12:52:11 -03:00
parent ee9a752d7f
commit e3566d1c21
5 changed files with 436 additions and 318 deletions
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.idea/.gitignore vendored
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# Datasource local storage ignored files # Datasource local storage ignored files
/dataSources/ /dataSources/
/dataSources.local.xml /dataSources.local.xml
# GitHub Copilot persisted chat sessions
/copilot/chatSessions

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.idea/casio-calculator.iml
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<?xml version="1.0" encoding="UTF-8"?> <?xml version="1.0" encoding="UTF-8"?>
<module type="PYTHON_MODULE" version="4"> <module type="PYTHON_MODULE" version="4">
<component name="NewModuleRootManager"> <component name="NewModuleRootManager">
<content url="file://$MODULE_DIR$" /> <content url="file://$MODULE_DIR$">
<excludeFolder url="file://$MODULE_DIR$/.idea/copilot/chatSessions" />
</content>
<orderEntry type="jdk" jdkName="Pipenv (casio-calculator)" jdkType="Python SDK" /> <orderEntry type="jdk" jdkName="Pipenv (casio-calculator)" jdkType="Python SDK" />
<orderEntry type="sourceFolder" forTests="false" /> <orderEntry type="sourceFolder" forTests="false" />
</component> </component>

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Pipfile
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@ -9,4 +9,3 @@ name = "pypi"
[requires] [requires]
python_version = "3.11" python_version = "3.11"
python_full_version = "3.11.7"

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Pipfile.lock generated
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{ {
"_meta": { "_meta": {
"hash": { "hash": {
"sha256": "bc82cd27f07d4e24b750064464bbc233a141778868b9a387125705e2d4e8a830" "sha256": "ed6d5d614626ae28e274e453164affb26694755170ccab3aa5866f093d51d3e4"
}, },
"pipfile-spec": 6, "pipfile-spec": 6,
"requires": { "requires": {
"python_full_version": "3.11.7",
"python_version": "3.11" "python_version": "3.11"
}, },
"sources": [ "sources": [

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distribution.py
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import math import math
import statistics
def factorial(n): def factorial(n):
""" """
Computes the factorial of a number. Computes the factorial of a number.
:param n: The number to compute the factorial of. :param n: The number to compute the factorial of.
:return: Returns the factorial of the number. :return: Returns the factorial of the number.
""" """
if n == 0: if n == 0:
return 1 return 1
for i in range(1, n): for i in range(1, n):
n *= i n *= i
return n return n
def combination(n, r): def combination(n, r):
""" """
Computes the combination of n choose r. Computes the combination of n choose r.
:param n: The number of items. :param n: The number of items.
:param r: The number of items to choose. :param r: The number of items to choose.
:return: Returns the number of ways to choose r items from n items. :return: Returns the number of ways to choose r items from n items.
""" """
return factorial(n) / (factorial(r) * factorial(n - r)) return factorial(n) / (factorial(r) * factorial(n - r))
def bnd(x, n, p): def bnd(x, n, p):
""" """
Computes the binomial distribution. Computes the binomial distribution.
:param x: Number of successes. :param x: Number of successes.
:param n: Number of trials. :param n: Number of trials.
:param p: Probability of success. :param p: Probability of success.
:return: Returns the probability of getting x successes in n trials. :return: Returns the probability of getting x successes in n trials.
""" """
return combination(n, x) * p ** x * (1 - p) ** (n - x) return combination(n, x) * p ** x * (1 - p) ** (n - x)
def bnd_mean(n, p): def bnd_mean(n, p):
""" """
Computes the mean of the binomial distribution. Computes the mean of the binomial distribution.
:param n: Number of trials. :param n: Number of trials.
:param p: Probability of success. :param p: Probability of success.
:return: Returns the mean of the binomial distribution. :return: Returns the mean of the binomial distribution.
""" """
return n * p return n * p
def bnd_var(n, p): def bnd_var(n, p):
""" """
Computes the variance of the binomial distribution. Computes the variance of the binomial distribution.
:param n: Number of trials. :param n: Number of trials.
:param p: Probability of success. :param p: Probability of success.
:return: Returns the variance of the binomial distribution. :return: Returns the variance of the binomial distribution.
""" """
return n * p * (1 - p) return n * p * (1 - p)
def bnd_std(n, p): def bnd_std(n, p):
""" """
Computes the standard deviation of the binomial distribution. Computes the standard deviation of the binomial distribution.
:param n: Number of trials. :param n: Number of trials.
:param p: Probability of success. :param p: Probability of success.
:return: Returns the standard deviation of the binomial distribution. :return: Returns the standard deviation of the binomial distribution.
""" """
return bnd_var(n, p) ** 0.5 return bnd_var(n, p) ** 0.5
def bnd_leq(x, n, p): def bnd_leq(x, n, p):
""" """
Computes the cumulative probability less than or equal to x successes in n trials. Computes the cumulative probability less than or equal to x successes in n trials.
:param x: Number of successes. :param x: Number of successes.
:param n: Number of trials. :param n: Number of trials.
:param p: Probability of success. :param p: Probability of success.
:return: Returns the cumulative probability less than or equal to x successes in n trials. :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)) return sum(bnd(i, n, p) for i in range(x + 1))
def bnd_lt(x, n, p): def bnd_lt(x, n, p):
""" """
Computes the cumulative probability less than x successes in n trials. Computes the cumulative probability less than x successes in n trials.
:param x: Number of successes. :param x: Number of successes.
:param n: Number of trials. :param n: Number of trials.
:param p: Probability of success. :param p: Probability of success.
:return: Returns the cumulative probability less than x successes in n trials. :return: Returns the cumulative probability less than x successes in n trials.
""" """
return sum(bnd(i, n, p) for i in range(x)) return sum(bnd(i, n, p) for i in range(x))
def bnd_geq(x, n, p): def bnd_geq(x, n, p):
""" """
Computes the cumulative probability greater than or equal to x successes in n trials. Computes the cumulative probability greater than or equal to x successes in n trials.
:param x: Number of successes. :param x: Number of successes.
:param n: Number of trials. :param n: Number of trials.
:param p: Probability of success. :param p: Probability of success.
:return: Returns the cumulative probability greater than or equal to x successes in n trials. :return: Returns the cumulative probability greater than or equal to x successes in n trials.
""" """
return 1 - bnd_lt(x, n, p) return 1 - bnd_lt(x, n, p)
def bnd_gt(x, n, p): def bnd_gt(x, n, p):
""" """
Computes the cumulative probability greater than x successes in n trials. Computes the cumulative probability greater than x successes in n trials.
:param x: Number of successes. :param x: Number of successes.
:param n: Number of trials. :param n: Number of trials.
:param p: Probability of success. :param p: Probability of success.
:return: Returns the cumulative probability greater than x successes in n trials. :return: Returns the cumulative probability greater than x successes in n trials.
""" """
return 1 - bnd_leq(x, n, p) return 1 - bnd_leq(x, n, p)
def gd(x, p, q=None): def gd(x, p, q=None):
""" """
Computes the geometric distribution. Computes the geometric distribution.
:param x: Number of trials until the first success. :param x: Number of trials until the first success.
:param p: Probability of success. :param p: Probability of success.
:param q: Probability of failure. :param q: Probability of failure.
:return: Returns the probability of getting the first success on the xth trial. :return: Returns the probability of getting the first success on the xth trial.
""" """
if q is None: if q is None:
q = 1 - p q = 1 - p
return q ** (x - 1) * p return q ** (x - 1) * p
def gd_mean(p): def gd_mean(p):
""" """
Computes the mean of the geometric distribution. Computes the mean of the geometric distribution.
:param p: Probability of success. :param p: Probability of success.
:return: Returns the mean of the geometric distribution. :return: Returns the mean of the geometric distribution.
""" """
return 1 / p return 1 / p
def gd_var(p): def gd_var(p):
""" """
Computes the variance of the geometric distribution. Computes the variance of the geometric distribution.
:param p: Probability of success. :param p: Probability of success.
:return: Returns the variance of the geometric distribution. :return: Returns the variance of the geometric distribution.
""" """
return (1 - p) / p ** 2 return (1 - p) / p ** 2
def gd_std(p): def gd_std(p):
""" """
Computes the standard deviation of the geometric distribution. Computes the standard deviation of the geometric distribution.
:param p: Probability of success. :param p: Probability of success.
:return: Returns the standard deviation of the geometric distribution. :return: Returns the standard deviation of the geometric distribution.
""" """
return gd_var(p) ** 0.5 return gd_var(p) ** 0.5
def gd_leq(x, p, q=None): def gd_leq(x, p, q=None):
""" """
Computes the cumulative probability of getting upto x trials until the first success. Computes the cumulative probability of getting upto x trials until the first success.
:param x: Number of trials until the first success. :param x: Number of trials until the first success.
:param p: Probability of success. :param p: Probability of success.
:param q: Probability of failure. :param q: Probability of failure.
:return: Returns the cumulative probability of getting upto x trials until the first success. :return: Returns the cumulative probability of getting upto x trials until the first success.
""" """
if q is not None: if q is not None:
return sum(gd(i, p, q) for i in range(1, x + 1)) 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)) return sum(gd(i, p) for i in range(1, x + 1))
def gd_lt(x, p, q=None): def gd_lt(x, p, q=None):
""" """
Computes the cumulative probability of getting less than x trials until the first success. Computes the cumulative probability of getting less than x trials until the first success.
:param x: Number of trials until the first success. :param x: Number of trials until the first success.
:param p: Probability of success. :param p: Probability of success.
:param q: Probability of failure. :param q: Probability of failure.
:return: Returns the cumulative probability of getting less than x trials until the first success. :return: Returns the cumulative probability of getting less than x trials until the first success.
""" """
if q is not None: if q is not None:
return sum(gd(i, p, q) for i in range(1, x)) return sum(gd(i, p, q) for i in range(1, x))
return sum(gd(i, p) for i in range(1, x)) return sum(gd(i, p) for i in range(1, x))
def gd_geq(x, p, q=None): def gd_geq(x, p, q=None):
""" """
Computes the cumulative probability of getting from x trials until the first success. Computes the cumulative probability of getting from x trials until the first success.
:param x: Number of trials until the first success. :param x: Number of trials until the first success.
:param p: Probability of success. :param p: Probability of success.
:param q: Probability of failure. :param q: Probability of failure.
:return: Returns the cumulative probability of getting from x trials until the first success. :return: Returns the cumulative probability of getting from x trials until the first success.
""" """
if q is not None: if q is not None:
return 1 - gd_lt(x, p, q) return 1 - gd_lt(x, p, q)
return 1 - gd_leq(x, p) return 1 - gd_leq(x, p)
def gd_gt(x, p, q=None): def gd_gt(x, p, q=None):
""" """
Computes the cumulative probability of getting from x trials until the first success. Computes the cumulative probability of getting from x trials until the first success.
:param x: Number of trials until the first success. :param x: Number of trials until the first success.
:param p: Probability of success. :param p: Probability of success.
:param q: Probability of failure. :param q: Probability of failure.
:return: Returns the cumulative probability of getting from x trials until the first success. :return: Returns the cumulative probability of getting from x trials until the first success.
""" """
if q is not None: if q is not None:
return 1 - gd_leq(x, p, q) return 1 - gd_leq(x, p, q)
return 1 - gd_leq(x, p) return 1 - gd_leq(x, p)
def hgd(x, N, n, k): def hgd(x, N, n, k):
""" """
Computes the hyper geometric distribution. Computes the hyper geometric distribution.
:param x: Number of successes in the sample. :param x: Number of successes in the sample.
:param N: Number of items in the population. :param N: Number of items in the population.
:param n: Number of draws. :param n: Number of draws.
:param k: Number of successes in the population. :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: 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) return (combination(k, x) * combination(N - k, n - x)) / combination(N, n)
def hgd_mean(N, n, k): def hgd_mean(N, n, k):
""" """
Computes the mean of the hyper geometric distribution. Computes the mean of the hyper geometric distribution.
:param N: Number of items in the population. :param N: Number of items in the population.
:param n: Number of draws. :param n: Number of draws.
:param k: Number of successes in the population. :param k: Number of successes in the population.
:return: Returns the mean of the hyper geometric distribution. :return: Returns the mean of the hyper geometric distribution.
""" """
return n * (k / N) return n * (k / N)
def hgd_var(N, n, k): def hgd_var(N, n, k):
""" """
Computes the variance of the hyper geometric distribution. Computes the variance of the hyper geometric distribution.
:param N: Number of items in the population. :param N: Number of items in the population.
:param n: Number of draws. :param n: Number of draws.
:param k: Number of successes in the population. :param k: Number of successes in the population.
:return: Returns the variance of the hyper geometric distribution. :return: Returns the variance of the hyper geometric distribution.
""" """
return (n * k * (N - k) * (N - n)) / (N ** 2 * (N - 1)) return (n * k * (N - k) * (N - n)) / (N ** 2 * (N - 1))
def hgd_std(N, n, k): def hgd_std(N, n, k):
""" """
Computes the standard deviation of the hyper geometric distribution. Computes the standard deviation of the hyper geometric distribution.
:param N: Number of items in the population. :param N: Number of items in the population.
:param n: Number of draws. :param n: Number of draws.
:param k: Number of successes in the population. :param k: Number of successes in the population.
:return: Returns the standard deviation of the hyper geometric distribution. :return: Returns the standard deviation of the hyper geometric distribution.
""" """
return hgd_var(N, n, k) ** 0.5 return hgd_var(N, n, k) ** 0.5
def hgd_leq(x, N, n, k): 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. 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 x: Number of successes in the sample.
:param N: Number of items in the population. :param N: Number of items in the population.
:param n: Number of draws. :param n: Number of draws.
:param k: Number of successes in the population. :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: 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)) return sum(hgd(i, N, n, k) for i in range(x + 1))
def hgd_lt(x, N, n, k): 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. 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 x: Number of successes in the sample.
:param N: Number of items in the population. :param N: Number of items in the population.
:param n: Number of draws. :param n: Number of draws.
:param k: Number of successes in the population. :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: 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)) return sum(hgd(i, N, n, k) for i in range(x))
def hgd_geq(x, N, n, k): 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. 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 x: Number of successes in the sample.
:param N: Number of items in the population. :param N: Number of items in the population.
:param n: Number of draws. :param n: Number of draws.
:param k: Number of successes in the population. :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: 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) return 1 - hgd_lt(x, N, n, k)
def hgd_gt(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. 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 x: Number of successes in the sample.
:param N: Number of items in the population. :param N: Number of items in the population.
:param n: Number of draws. :param n: Number of draws.
:param k: Number of successes in the population. :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: 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) return 1 - hgd_leq(x, N, n, k)
def pd(x, l): def pd(x, l):
""" """
Computes the poisson distribution. Computes the poisson distribution.
:param x: Number of occurrences. :param x: Number of occurrences.
:param l: Average number of occurrences. :param l: Average number of occurrences.
:return: Returns the probability of getting x occurrences. :return: Returns the probability of getting x occurrences.
""" """
return (l ** x * math.e ** -l) / factorial(x) return (l ** x * math.e ** -l) / factorial(x)
def pd_mean(l): def pd_mean(l):
""" """
Computes the mean of the poisson distribution. Computes the mean of the poisson distribution.
:param l: Average number of occurrences. :param l: Average number of occurrences.
:return: Returns the mean of the poisson distribution. :return: Returns the mean of the poisson distribution.
""" """
return l return l
def pd_var(l): def pd_var(l):
""" """
Computes the variance of the poisson distribution. Computes the variance of the poisson distribution.
:param l: Average number of occurrences. :param l: Average number of occurrences.
:return: Returns the variance of the poisson distribution. :return: Returns the variance of the poisson distribution.
""" """
return l return l
def pd_std(l): def pd_std(l):
""" """
Computes the standard deviation of the poisson distribution. Computes the standard deviation of the poisson distribution.
:param l: Average number of occurrences. :param l: Average number of occurrences.
:return: Returns the standard deviation of the poisson distribution. :return: Returns the standard deviation of the poisson distribution.
""" """
return l ** 0.5 return l ** 0.5
def pd_leq(x, l): def pd_leq(x, l):
""" """
Computes the cumulative probability of getting upto x occurrences. Computes the cumulative probability of getting upto x occurrences.
:param x: Number of occurrences. :param x: Number of occurrences.
:param l: Average number of occurrences. :param l: Average number of occurrences.
:return: Returns the cumulative probability of getting upto x occurrences. :return: Returns the cumulative probability of getting upto x occurrences.
""" """
return sum(pd(i, l) for i in range(x + 1)) return sum(pd(i, l) for i in range(x + 1))
def pd_lt(x, l): def pd_lt(x, l):
""" """
Computes the cumulative probability of getting less than x occurrences. Computes the cumulative probability of getting less than x occurrences.
:param x: Number of occurrences. :param x: Number of occurrences.
:param l: Average number of occurrences. :param l: Average number of occurrences.
:return: Returns the cumulative probability of getting less than x occurrences. :return: Returns the cumulative probability of getting less than x occurrences.
""" """
return sum(pd(i, l) for i in range(x)) return sum(pd(i, l) for i in range(x))
def pd_geq(x, l): def pd_geq(x, l):
""" """
Computes the cumulative probability of getting from x occurrences. Computes the cumulative probability of getting from x occurrences.
:param x: Number of occurrences. :param x: Number of occurrences.
:param l: Average number of occurrences. :param l: Average number of occurrences.
:return: Returns the cumulative probability of getting from x occurrences. :return: Returns the cumulative probability of getting from x occurrences.
""" """
return 1 - pd_lt(x, l) return 1 - pd_lt(x, l)
def pd_gt(x, l): def pd_gt(x, l):
""" """
Computes the cumulative probability of getting from x occurrences. Computes the cumulative probability of getting from x occurrences.
:param x: Number of occurrences. :param x: Number of occurrences.
:param l: Average number of occurrences. :param l: Average number of occurrences.
:return: Returns the cumulative probability of getting from x occurrences. :return: Returns the cumulative probability of getting from x occurrences.
""" """
return 1 - pd_leq(x, l) 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(): def man():
""" """
Prints the manual for the module. Prints the manual for the module.
""" """
seperator = "-" * 20 separator = "-" * 20
print("This module contains functions for computing the total probability of events.") print("This module contains functions for computing the total probability of events.")
print("The functions are:") print("The functions are:")
print(seperator) print(separator)
print("Binomial Distribution") print("Binomial Distribution")
print("bnd(x, n, p)") print("bnd(x, n, p)")
print("bnd_mean(n, p)") print("bnd_mean(n, p)")
print("bnd_var(n, p)") print("bnd_var(n, p)")
print("bnd_std(n, p)") print("bnd_std(n, p)")
print("bnd_leq(x, n, p)") print("bnd_leq(x, n, p)")
print("bnd_lt(x, n, p)") print("bnd_lt(x, n, p)")
print("bnd_geq(x, n, p)") print("bnd_geq(x, n, p)")
print("bnd_gt(x, n, p)") print("bnd_gt(x, n, p)")
print(seperator) print(separator)
print("Geometric Distribution") print("Geometric Distribution")
print("gd(x, p, q)") print("gd(x, p, q)")
print("gd_mean(p)") print("gd_mean(p)")
print("gd_var(p)") print("gd_var(p)")
print("gd_std(p)") print("gd_std(p)")
print("gd_leq(x, p, q)") print("gd_leq(x, p, q)")
print("gd_lt(x, p, q)") print("gd_lt(x, p, q)")
print("gd_geq(x, p, q)") print("gd_geq(x, p, q)")
print("gd_gt(x, p, q)") print("gd_gt(x, p, q)")
print(seperator) print(separator)
print("Hyper Geometric Distribution") print("Hyper Geometric Distribution")
print("hgd(x, N, n, k)") print("hgd(x, N, n, k)")
print("hgd_mean(N, n, k)") print("hgd_mean(N, n, k)")
print("hgd_var(N, n, k)") print("hgd_var(N, n, k)")
print("hgd_std(N, n, k)") print("hgd_std(N, n, k)")
print("hgd_leq(x, N, n, k)") print("hgd_leq(x, N, n, k)")
print("hgd_lt(x, N, n, k)") print("hgd_lt(x, N, n, k)")
print("hgd_geq(x, N, n, k)") print("hgd_geq(x, N, n, k)")
print("hgd_gt(x, N, n, k)") print("hgd_gt(x, N, n, k)")
print(seperator) print(separator)
print("Poisson Distribution") print("Poisson Distribution")
print("pd(x, l)") print("pd(x, l)")
print("pd_mean(l)") print("pd_mean(l)")
print("pd_var(l)") print("pd_var(l)")
print("pd_std(l)") print("pd_std(l)")
print("pd_leq(x, l)") print("pd_leq(x, l)")
print("pd_lt(x, l)") print("pd_lt(x, l)")
print("pd_geq(x, l)") print("pd_geq(x, l)")
print("pd_gt(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)")