Converting a Uniform Distribution to a Normal Dist

Question

How can I convert a uniform distribution (as most random number generators produce, e.g. between 0.0 and 1.0) into a normal distribution? What if I want a mean and standard deviation of my choosing?

Answer 1

It seems incredible that I could add something to this after eight years, but for the case of Java I would like to point readers to the Random.nextGaussian() method, which generates a Gaussian distribution with mean 0.0 and standard deviation 1.0 for you.

A simple addition and/or multiplication will change the mean and standard deviation to your needs.

Answer 2

Where R1, R2 are random uniform numbers:

NORMAL DISTRIBUTION, with SD of 1: sqrt(-2*log(R1))*cos(2*pi*R2)

This is exact... no need to do all those slow loops!

Answer 3

Here is a javascript implementation using the polar form of the Box-Muller transformation.

/*
 * Returns member of set with a given mean and standard deviation
 * mean: mean
 * standard deviation: std_dev 
 */
function createMemberInNormalDistribution(mean,std_dev){
    return mean + (gaussRandom()*std_dev);
}

/*
 * Returns random number in normal distribution centering on 0.
 * ~95% of numbers returned should fall between -2 and 2
 * ie within two standard deviations
 */
function gaussRandom() {
    var u = 2*Math.random()-1;
    var v = 2*Math.random()-1;
    var r = u*u + v*v;
    /*if outside interval [0,1] start over*/
    if(r == 0 || r >= 1) return gaussRandom();

    var c = Math.sqrt(-2*Math.log(r)/r);
    return u*c;

    /* todo: optimize this algorithm by caching (v*c) 
     * and returning next time gaussRandom() is called.
     * left out for simplicity */
}

Answer 4

There are plenty of methods:

• Do not use Box Muller. Especially if you draw many gaussian numbers. Box Muller yields a result which is clamped between -6 and 6 (assuming double precision. Things worsen with floats.). And it is really less efficient than other available methods.
• Ziggurat is fine, but needs a table lookup (and some platform-specific tweaking due to cache size issues)
• Ratio-of-uniforms is my favorite, only a few addition/multiplications and a log 1/50th of the time (eg. look there).
• Inverting the CDF is efficient (and overlooked, why ?), you have fast implementations of it available if you search google. It is mandatory for Quasi-Random numbers.

Answer 5

I thing you should try this in EXCEL: =norminv(rand();0;1). This will product the random numbers which should be normally distributed with the zero mean and unite variance. "0" can be supplied with any value, so that the numbers will be of desired mean, and by changing "1", you will get the variance equal to the square of your input.

For example: =norminv(rand();50;3) will yield to the normally distributed numbers with MEAN = 50 VARIANCE = 9.

Answer 6

Changing the distribution of any function to another involves using the inverse of the function you want.

In other words, if you aim for a specific probability function p(x) you get the distribution by integrating over it -> d(x) = integral(p(x)) and use its inverse: Inv(d(x)). Now use the random probability function (which have uniform distribution) and cast the result value through the function Inv(d(x)). You should get random values cast with distribution according to the function you chose.

This is the generic math approach - by using it you can now choose any probability or distribution function you have as long as it have inverse or good inverse approximation.

Hope this helped and thanks for the small remark about using the distribution and not the probability itself.