Combine n Gaussian Probability Sets in a limited float range

Question

I've got a fixed solution space defined by a minimum and maximum float which has no divisible slice.

You then have 0..N Gaussian distribution probability sets which I need to combine.

In the end I neeed

1. Method to define probability set with a numeric range (not sliceable) and 0..N Gaussian functions
2. A function which can generate a random number in the range as defined by the calculated probabilities.

Also I know it is possible that some combinations will generate a zero solution space.

Now I'm thinking the way to do it is take the normalised probability density functions and multiply them to get a new space then normalising the result. I just can't seem to break it down into algorithmic form.

Any ideas?

Answer 1

First let's make sure I understood your question correctly:

• You have a probability function that is expressed as a sum of N parametrized one-dimensional gaussians, each with different mean and standard deviation.
• You want to generate stochastic variables according to this distribution.

Is this correct?

If this is the case, I reccommend you use a variation of rejection sampling. The recipe is quite straightforward, but you might have to iterate a bit before you get an answer out of it. This is the basic outline.

1. You generate a uniformly distributed random number in your desired interval, x.
2. You calculate the value of your probability distribution, p(x)
3. You generate another uniformly distributed random number between 0 and 1, q
4. If q < p(x), return x
5. If not, start from step 1.

No matter how large the temptation, do not re-use q for different iterations.