2024-02-17 19:18:01 -04:00
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import math
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2024-02-17 22:09:00 -04:00
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def factorial(n):
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"""
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Computes the factorial of a number.
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:param n: The number to compute the factorial of.
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:return: Returns the factorial of the number.
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"""
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if n == 0:
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return 1
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for i in range(1, n):
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n *= i
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return n
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def combination(n, r):
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"""
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Computes the combination of n choose r.
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:param n: The number of items.
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:param r: The number of items to choose.
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:return: Returns the number of ways to choose r items from n items.
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"""
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return factorial(n) / (factorial(r) * factorial(n - r))
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2024-02-17 19:18:01 -04:00
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def bnd(x, n, p):
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"""
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Computes the binomial distribution.
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:param x: Number of successes.
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:param n: Number of trials.
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:param p: Probability of success.
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:return: Returns the probability of getting x successes in n trials.
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"""
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2024-02-17 22:09:00 -04:00
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return combination(n, x) * p ** x * (1 - p) ** (n - x)
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2024-02-17 19:18:01 -04:00
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def bnd_mean(n, p):
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"""
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Computes the mean of the binomial distribution.
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:param n: Number of trials.
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:param p: Probability of success.
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:return: Returns the mean of the binomial distribution.
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"""
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return n * p
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def bnd_var(n, p):
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"""
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Computes the variance of the binomial distribution.
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:param n: Number of trials.
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:param p: Probability of success.
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:return: Returns the variance of the binomial distribution.
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"""
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return n * p * (1 - p)
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def bnd_std(n, p):
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"""
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Computes the standard deviation of the binomial distribution.
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:param n: Number of trials.
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:param p: Probability of success.
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:return: Returns the standard deviation of the binomial distribution.
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"""
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return bnd_var(n, p) ** 0.5
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2024-02-17 21:44:21 -04:00
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def bnd_leq(x, n, p):
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2024-02-17 19:18:01 -04:00
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"""
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2024-02-17 21:44:21 -04:00
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Computes the cumulative probability less than or equal to x successes in n trials.
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2024-02-17 19:18:01 -04:00
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:param x: Number of successes.
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:param n: Number of trials.
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:param p: Probability of success.
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2024-02-17 21:44:21 -04:00
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:return: Returns the cumulative probability less than or equal to x successes in n trials.
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2024-02-17 19:18:01 -04:00
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"""
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return sum(bnd(i, n, p) for i in range(x + 1))
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2024-02-17 21:44:21 -04:00
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def bnd_lt(x, n, p):
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2024-02-17 19:18:01 -04:00
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"""
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2024-02-17 21:44:21 -04:00
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Computes the cumulative probability less than x successes in n trials.
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2024-02-17 19:18:01 -04:00
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:param x: Number of successes.
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:param n: Number of trials.
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:param p: Probability of success.
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2024-02-17 21:44:21 -04:00
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:return: Returns the cumulative probability less than x successes in n trials.
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2024-02-17 19:18:01 -04:00
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"""
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2024-02-17 21:44:21 -04:00
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return sum(bnd(i, n, p) for i in range(x))
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def bnd_geq(x, n, p):
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"""
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Computes the cumulative probability greater than or equal to x successes in n trials.
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:param x: Number of successes.
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:param n: Number of trials.
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:param p: Probability of success.
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:return: Returns the cumulative probability greater than or equal to x successes in n trials.
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"""
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return 1 - bnd_lt(x, n, p)
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def bnd_gt(x, n, p):
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"""
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Computes the cumulative probability greater than x successes in n trials.
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:param x: Number of successes.
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:param n: Number of trials.
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:param p: Probability of success.
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:return: Returns the cumulative probability greater than x successes in n trials.
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"""
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return 1 - bnd_leq(x, n, p)
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2024-02-17 19:18:01 -04:00
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def gd(x, p, q=None):
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"""
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Computes the geometric distribution.
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:param x: Number of trials until the first success.
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:param p: Probability of success.
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:param q: Probability of failure.
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:return: Returns the probability of getting the first success on the xth trial.
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"""
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if q is None:
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q = 1 - p
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return q ** (x - 1) * p
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def gd_mean(p):
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"""
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Computes the mean of the geometric distribution.
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:param p: Probability of success.
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:return: Returns the mean of the geometric distribution.
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"""
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return 1 / p
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def gd_var(p):
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"""
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Computes the variance of the geometric distribution.
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:param p: Probability of success.
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:return: Returns the variance of the geometric distribution.
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"""
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return (1 - p) / p ** 2
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def gd_std(p):
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"""
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Computes the standard deviation of the geometric distribution.
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:param p: Probability of success.
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:return: Returns the standard deviation of the geometric distribution.
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"""
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return gd_var(p) ** 0.5
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2024-02-17 21:44:21 -04:00
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def gd_leq(x, p, q=None):
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2024-02-17 19:18:01 -04:00
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"""
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Computes the cumulative probability of getting upto x trials until the first success.
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:param x: Number of trials until the first success.
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:param p: Probability of success.
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2024-02-17 19:50:41 -04:00
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:param q: Probability of failure.
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2024-02-17 19:18:01 -04:00
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:return: Returns the cumulative probability of getting upto x trials until the first success.
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"""
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2024-02-17 19:50:41 -04:00
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if q is not None:
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return sum(gd(i, p, q) for i in range(1, x + 1))
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2024-02-17 19:18:01 -04:00
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return sum(gd(i, p) for i in range(1, x + 1))
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2024-02-17 21:44:21 -04:00
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def gd_lt(x, p, q=None):
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"""
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Computes the cumulative probability of getting less than x trials until the first success.
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:param x: Number of trials until the first success.
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:param p: Probability of success.
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:param q: Probability of failure.
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:return: Returns the cumulative probability of getting less than x trials until the first success.
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"""
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if q is not None:
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return sum(gd(i, p, q) for i in range(1, x))
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return sum(gd(i, p) for i in range(1, x))
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def gd_geq(x, p, q=None):
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"""
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Computes the cumulative probability of getting from x trials until the first success.
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:param x: Number of trials until the first success.
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:param p: Probability of success.
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:param q: Probability of failure.
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:return: Returns the cumulative probability of getting from x trials until the first success.
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"""
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if q is not None:
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return 1 - gd_lt(x, p, q)
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return 1 - gd_leq(x, p)
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def gd_gt(x, p, q=None):
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2024-02-17 19:18:01 -04:00
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"""
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Computes the cumulative probability of getting from x trials until the first success.
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:param x: Number of trials until the first success.
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:param p: Probability of success.
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2024-02-17 19:50:41 -04:00
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:param q: Probability of failure.
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2024-02-17 19:18:01 -04:00
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:return: Returns the cumulative probability of getting from x trials until the first success.
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"""
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2024-02-17 19:50:41 -04:00
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if q is not None:
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2024-02-17 21:44:21 -04:00
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return 1 - gd_leq(x, p, q)
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return 1 - gd_leq(x, p)
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2024-02-17 19:18:01 -04:00
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def hgd(x, N, n, k):
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"""
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Computes the hyper geometric distribution.
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:param x: Number of successes in the sample.
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:param N: Number of items in the population.
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:param n: Number of draws.
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:param k: Number of successes in the population.
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:return: Returns the probability of getting x successes in n draws from a population of size N with k successes.
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"""
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2024-02-17 22:09:00 -04:00
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return (combination(k, x) * combination(N - k, n - x)) / combination(N, n)
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2024-02-17 19:18:01 -04:00
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def hgd_mean(N, n, k):
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"""
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Computes the mean of the hyper geometric distribution.
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:param N: Number of items in the population.
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:param n: Number of draws.
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:param k: Number of successes in the population.
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:return: Returns the mean of the hyper geometric distribution.
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"""
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return n * (k / N)
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def hgd_var(N, n, k):
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"""
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Computes the variance of the hyper geometric distribution.
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:param N: Number of items in the population.
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:param n: Number of draws.
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:param k: Number of successes in the population.
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:return: Returns the variance of the hyper geometric distribution.
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"""
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return (n * k * (N - k) * (N - n)) / (N ** 2 * (N - 1))
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def hgd_std(N, n, k):
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"""
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Computes the standard deviation of the hyper geometric distribution.
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:param N: Number of items in the population.
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:param n: Number of draws.
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:param k: Number of successes in the population.
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:return: Returns the standard deviation of the hyper geometric distribution.
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"""
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return hgd_var(N, n, k) ** 0.5
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2024-02-17 21:44:21 -04:00
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def hgd_leq(x, N, n, k):
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2024-02-17 19:18:01 -04:00
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"""
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Computes the cumulative probability of getting upto x successes in n draws from a population of size N with k successes.
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:param x: Number of successes in the sample.
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:param N: Number of items in the population.
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:param n: Number of draws.
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:param k: Number of successes in the population.
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:return: Returns the cumulative probability of getting upto x successes in n draws from a population of size N with k successes.
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"""
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return sum(hgd(i, N, n, k) for i in range(x + 1))
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2024-02-17 21:44:21 -04:00
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def hgd_lt(x, N, n, k):
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"""
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Computes the cumulative probability of getting less than x successes in n draws from a population of size N with k successes.
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:param x: Number of successes in the sample.
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:param N: Number of items in the population.
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:param n: Number of draws.
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:param k: Number of successes in the population.
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:return: Returns the cumulative probability of getting less than x successes in n draws from a population of size N with k successes.
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"""
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return sum(hgd(i, N, n, k) for i in range(x))
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def hgd_geq(x, N, n, k):
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"""
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Computes the cumulative probability of getting from x successes in n draws from a population of size N with k successes.
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:param x: Number of successes in the sample.
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:param N: Number of items in the population.
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:param n: Number of draws.
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:param k: Number of successes in the population.
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:return: Returns the cumulative probability of getting from x successes in n draws from a population of size N with k successes.
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"""
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return 1 - hgd_lt(x, N, n, k)
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def hgd_gt(x, N, n, k):
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2024-02-17 19:18:01 -04:00
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"""
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Computes the cumulative probability of getting from x successes in n draws from a population of size N with k successes.
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:param x: Number of successes in the sample.
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:param N: Number of items in the population.
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:param n: Number of draws.
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:param k: Number of successes in the population.
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:return: Returns the cumulative probability of getting from x successes in n draws from a population of size N with k successes.
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"""
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2024-02-17 21:44:21 -04:00
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return 1 - hgd_leq(x, N, n, k)
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2024-02-17 19:18:01 -04:00
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def pd(x, l):
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"""
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Computes the poisson distribution.
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:param x: Number of occurrences.
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:param l: Average number of occurrences.
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:return: Returns the probability of getting x occurrences.
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"""
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return (l ** x * math.e ** -l) / factorial(x)
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def pd_mean(l):
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"""
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Computes the mean of the poisson distribution.
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:param l: Average number of occurrences.
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:return: Returns the mean of the poisson distribution.
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"""
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return l
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def pd_var(l):
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"""
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Computes the variance of the poisson distribution.
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:param l: Average number of occurrences.
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:return: Returns the variance of the poisson distribution.
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"""
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return l
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def pd_std(l):
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"""
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Computes the standard deviation of the poisson distribution.
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:param l: Average number of occurrences.
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:return: Returns the standard deviation of the poisson distribution.
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"""
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return l ** 0.5
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def pd_leq(x, l):
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"""
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Computes the cumulative probability of getting upto x occurrences.
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:param x: Number of occurrences.
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:param l: Average number of occurrences.
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:return: Returns the cumulative probability of getting upto x occurrences.
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"""
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return sum(pd(i, l) for i in range(x + 1))
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2024-02-17 21:44:21 -04:00
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def pd_lt(x, l):
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"""
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Computes the cumulative probability of getting less than x occurrences.
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:param x: Number of occurrences.
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:param l: Average number of occurrences.
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:return: Returns the cumulative probability of getting less than x occurrences.
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"""
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return sum(pd(i, l) for i in range(x))
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def pd_geq(x, l):
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"""
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Computes the cumulative probability of getting from x occurrences.
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:param x: Number of occurrences.
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:param l: Average number of occurrences.
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:return: Returns the cumulative probability of getting from x occurrences.
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"""
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return 1 - pd_lt(x, l)
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def pd_gt(x, l):
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2024-02-17 19:18:01 -04:00
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"""
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Computes the cumulative probability of getting from x occurrences.
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:param x: Number of occurrences.
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:param l: Average number of occurrences.
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:return: Returns the cumulative probability of getting from x occurrences.
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"""
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2024-02-17 21:44:21 -04:00
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return 1 - pd_leq(x, l)
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2024-02-17 19:18:01 -04:00
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2024-02-17 19:33:16 -04:00
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def man():
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2024-02-17 21:51:18 -04:00
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seperator = "-" * 20
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2024-02-17 19:50:41 -04:00
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"""
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Prints the manual for the module.
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2024-02-17 21:24:22 -04:00
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Formatted this way to fit in memory on the calculator.
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"""
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print("This module contains functions for computing the total probability of events.")
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print("The functions are:")
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2024-02-17 21:44:21 -04:00
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print(seperator)
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2024-02-17 21:51:18 -04:00
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print("Binomial Distribution")
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print("bnd(x, n, p)")
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print("bnd_mean(n, p)")
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print("bnd_var(n, p)")
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print("bnd_std(n, p)")
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print("bnd_leq(x, n, p)")
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print("bnd_lt(x, n, p)")
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print("bnd_geq(x, n, p)")
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print("bnd_gt(x, n, p)")
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2024-02-17 21:44:21 -04:00
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print(seperator)
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2024-02-17 21:51:18 -04:00
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print("Geometric Distribution")
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print("gd(x, p, q)")
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print("gd_mean(p)")
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print("gd_var(p)")
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print("gd_std(p)")
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print("gd_leq(x, p, q)")
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print("gd_lt(x, p, q)")
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print("gd_geq(x, p, q)")
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print("gd_gt(x, p, q)")
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2024-02-17 21:44:21 -04:00
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print(seperator)
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2024-02-17 21:51:18 -04:00
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print("Hyper Geometric Distribution")
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print("hgd(x, N, n, k)")
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print("hgd_mean(N, n, k)")
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print("hgd_var(N, n, k)")
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print("hgd_std(N, n, k)")
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print("hgd_leq(x, N, n, k)")
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print("hgd_lt(x, N, n, k)")
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print("hgd_geq(x, N, n, k)")
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print("hgd_gt(x, N, n, k)")
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print(seperator)
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print("Poisson Distribution")
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print("pd(x, l)")
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print("pd_mean(l)")
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print("pd_var(l)")
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print("pd_std(l)")
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print("pd_leq(x, l)")
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print("pd_lt(x, l)")
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print("pd_geq(x, l)")
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print("pd_gt(x, l)")
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