import numpy as np
from scipy.special import erfinv
import scipy.stats as stats
import math
[docs]
def qnorm(p, mean=0, sd=1, lower_tail=True):
"""
Quantile (Inverse Cumulative Distribution Function) of the normal distribution.
Parameters
----------
p: np.float64
The probability for which to find the quantile. Must be between 0 and 1 (exclusive).
mean: np.float64, optional
The mean (average) of the normal distribution. Default is 0.
sd: np.float64, optional
The standard deviation of the normal distribution. Default is 1.
lower_tail: bool, optional
If True, return the probability to the left of p in the normal distribution.
If False, return the probability to the right of p in the normal distribution.
Default is True.
Returns
-------
np.float64
Returns the value of the inverse cumulative density function (cdf) of the normal distribution
given a certain random variable p, a population mean μ, and the population standard deviation σ.
Formula
-------
If lower_tail=True:
Q(p; μ, σ) = μ + σ * sqrt(2) * erfinv(2p - 1)
If lower_tail=False:
Q(p; μ, σ) = μ - σ * sqrt(2) * erfinv(2p - 1)
The quantile represents the value below which a given proportion of the distribution
falls. It is characterized by the mean (`μ`) and standard deviation (`σ`), determining
the center and spread of the distribution.
Example
-------
>>> qnorm(0.8413447460685429, mean=0, sd=1, lower_tail=True)
1.0
"""
# Input type checks
if not all(isinstance(param, (int, float)) for param in [p, mean, sd]):
raise TypeError(f"Input parameters must be numerical")
if not isinstance(lower_tail, bool):
raise TypeError(f"Expected input to be boolean, got {type(lower_tail)}")
# Value checks
if not 0 < p < 1:
raise ValueError("p must be between 0 and 1 (exclusive).")
if not sd > 0:
raise ValueError("standard deviation must be positive")
if lower_tail:
result = mean + sd * np.sqrt(2) * erfinv(2 * p - 1)
return result
else:
result = mean - sd * np.sqrt(2) * erfinv(2 * p - 1)
return result