# Transform standard normally distributed numbers into arbitrary normal distribution

I need a random number generator with a normal distribution, with parameterizable mean and standard deviation. I have only a uniformly distributed random number generator. By applying a Box-Muller transform to two uniformly distributed numbers in [0,1), I get a standard normal distribution. However, I've become confused about how to get an arbitrary, parameterized normal distribution by this procedure.

Since you can use a z-transform to change a non-standard normal distribution into a standard one, I thought that I could just use the inverse of a z-transform, and pass the result of the Box-Muller-transformed uniformly distributed numbers into $(x \times \sigma) + \mu$, where $\sigma$ is the desired standard deviation and $\mu$ is the desired mean. I've become convinced that this is a mistake. My simulation, using the transformed RNG didn't behave properly (I'm able to compare it with a well-tested normally distributed RNG), and a plot of the transformed distribution doesn't look at all correct.

I think that what I need to do is to do something with the mean and standard deviation inside the Box-Muller code, rather than just processing its return value in a simple way. However, I'm confused about how to do this.

In case it's helpful, here is my Box-Muller code. (It's in the NetLogo language, which may not be familiar, but I think this bit code will be intelligible to anyone with programming experience.)

to-report box-muller-random-normal [mn stddev]
if-else box-muller-cache-exists = 0 [  ; using 1 for true and 0 for false means we don't have to initialize to false in setup; just use default zero init value
; cache is unfilled, so compute new pair of values
let x1 random-float 1
let x2 random-float 1
let y1 sqrt (-2 * ln x1) * cos (2 * pi * x2)
let y2 sqrt (-2 * ln x1) * sin (2 * pi * x2)
set box-muller-cache y2
set box-muller-cache-exists 1
report y1
][
set box-muller-cache-exists 0    ; so cache will be refilled next time
report box-muller-cache
]
end


Where would I insert transformations involving the mean and standard deviation? (to-report is NetLogo's function-definition command. The code using ...cache... variables is designed to remember one of the values between invocations, generating new numbers only if there is no remembered value.)

• why are you implementing box-muller yourself/by hand?? At a quick glance it looks like NetLogo has a built-in normal deviate generator ... unless you're doing something not in NetLogo but have chosen to express your algorithm in NetLogo code (which would be pretty weird ...) Sep 21, 2015 at 2:14
• Right, but NetLogo Web hasn't yet implemented it. I need to make a NetLogo model available to my students now, without requiring them to install NetLogo.
– Mars
Sep 21, 2015 at 2:18
• oh, OK. Makes sense. Sep 21, 2015 at 2:35

If $X \sim N(0,1)$, then $Y = \sigma X + \mu \sim N(\mu,\sigma^2)$. So, leave the RNG and Box-Muller alone, and when you get the generated variate, just multiply by $\sigma$ and add $\mu$. You could stick these inside the Box-Muller code if you really want just simple code by modifying
let y1 sqrt (2 - ln x1) * cos (2 * pi * x2)
let y1 sigma * sqrt (-2 * ln x1) * cos (2 * pi * x2) + mu
and similarly for y2.