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Box-Muller Transformation

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Prob. and Stats

Difficulty: 2 | Problem written by mesakarghm
Problem reported in interviews at

Apple

The Box-Muller Rransform is a method for generating normally distributed random numbers from uniformly distributed random numbers. The Box-Muller Transformation can be summarized as follows. 

Suppose u1 and u2 are independent random variables that are uniformly distributed between 0 and 1 and let:

\(z_1 = \sqrt {-2 log(u_1)} {cos (2\pi u_2)} \)

\(z_2 = \sqrt {-2 log(u_1)} {sin (2\pi u_2)} \)

Then, z1 and z2 are independent random variables with a standard normal distribution. 

Write a function transform(u1,u2) such that when given two uniformly distributed independent random variables (integers), it generates a tuple containing z1 and z2 with standard normal distribution using Box-Muller Transform.

Sample Input:
<class 'float'>
u1: 0.66971127
<class 'float'>
u2: 0.50960964

Expected Output:
<class 'tuple'>
(-0.8938107248822067, -0.05403320867504681)

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numpy has been already imported as np (import numpy as np)