Combine $n$ Normal distribution 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 Normal distribution probability sets which I need to combine.

In the end I neeed

Method to define probability set with a numeric range (not sliceable) and 0..N Gaussian functions 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?

Extra Clarification

Let's examine the following height rules.

STATUS Height [ 1.2 : 2.4 ] MODIFIER Gender[ Male ] {Height [ 1.9 ~ 0.4 ] } MODIFIER Nation[ Red ] { Height [ 1.7 ~ 0.2 ] }

Now assuming that we have a man from the Red nation we need to resolve these. We know that the original status declaration defines the entire possibility space. We cannot leave that space.

Now basically I need to find a way of combining these to get a new probability. What I meant by slice was because its a floating point number I can't break it into a set of elements of calculate the new probability for each element. I can't slice it up into pieces.

Answer 1

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

• You have a probability function that is expressed as a sum/product 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.

Answer 2

First up, you should accept Ranieri's answer, which is obviously correct. The question of "slicing" seems like a temporary misunderstanding that shouldn't stand in your way. Rejection sampling works with continuous distributions.

Second, it seems you are attempting to randomly generate individuals from a population with some number of subgroups, each of which has an expected distribution of some continuous traits. Since the traits are distributed normally, $\forall{x}: P(x) > 0$, so it is notionally possible to have outlier individuals with any trait combination. However, it is very likely that arbitrary distributional constraints will not be satisfiable at the population level.

For example, imagine that you have a population divided into two genders and two colours, and specified to be 50/50 male/female and 50/50 red/blue. Then require that some trait, say foolishness, is distributed $N(1,0.1)$ for each of the subgroups men, red and blue, but $N(0.5, 0.1)$ for women. Is this possible? I'm pretty sure not. And that's without even trying to clip values into a fixed range.

So, I suspect that you might need a somewhat better model for your probability distribution. My first inclination would be to weight the different criteria in some fashion to prioritise the distribution of some traits over others. However, I'd have to think that through properly, which I don't have time to do just now. Maybe later...