Direct Transformation for the Normal Distribution

In the previous section, a simple, but less accurate method of generating a normal distribution was presented. Here we consider a direct transormation.

• Let $Z_1, Z_2$ be two standard normal random variates.
• Plot the two as a point in the plane and represent them in a polar coordinate system as
  
  $$\begin{array}{ll} Z_1 = B \cos \theta \\ Z_2 = B \sin \theta \\ \end{array}$$
  
  

• It is known that $B^2 = {Z_1}^2 + {Z_2}^2$ has the chi-square distribution with 2 degrees of freedom, which is equivalent to an exponential distribution with mean 2
  
  $$Y = \lambda e^{-\lambda t}, ~~~~ t \ge 0$$
  
  
  
  
  $$E[Y] = 2 = \lambda$$
  
  
  thus the raidus B can be generated using (9.3)
  
  $$B = \sqrt{-2\ln R}$$
  
  

• So a normal distribution can be generated by any one of the following.
  
  $$Z_1 = \sqrt{-2 \ln R_1} \cos (2\pi R_2)$$
  
  
  
  
  $$Z_2 = \sqrt{-2 \ln R_1} \sin (2\pi R_2)$$
  
  
  where $R_1$ and $R_2$ are uniformly distributed over (0,1).

• To obtain normail variates $X_i$ with mean $\mu$ and variance $\sigma^2$, transform
  
  $$X_i = \mu + \sigma Z_i$$