Ways to generate normal distribution random variable
Nov 7, 2021
- directly use NumPy method
import numpy as nprv = np.random.normal(0,1)
2. use the central limit theorem
rv = (np.mean(np.random.uniform(0,1,n))-0.5)*np.sqrt(12*n)3. use inverse-CDF
from scipy.stats import normiid = np.random.uniform(0,1,size)rv = norm.ppf(iid)
quick proof of inverse-CDF correctness [1]:
Let F be a continuous cumulative distribution function, and let F^-1 be its inverse function
Claim: If U is a uniform random variable on (0, 1) then F^-1(U) has F as its CDF.
Proof:
Pr(F^-1(U) < x)
= Pr(F(x) > U) (apply F both side)
= F(x) (because of uniform distribution)
4. Box-Muller method [2]
x = np.random.uniform(0,1)y = np.random.uniform(0,1)rv1 = np.cos(2*pi*x)*np.sqrt(-2*np.log(1-y))rv2 = np.sin(2*pi*x)*np.sqrt(-2*np.log(1-y))
The key of Box-Muller is still inverse-CDF, but we consider a joint distribution of x, y
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