# Useful separation value in a mixture distribution

Assume we have a distribution that is the mixture of two normal distributions. The pdf of the overall distributions and their single parts may look like the following. In black, the combined distribution and colors are their parts

Sometimes it is necessary to obtain a certain separation value that lies between the two mean values of the parts. One idea I was playing with is to use the confidence intervals for this. For the blue distribution it is {80,120} and for the yellow one we get {111, 189}. Now the one can use the mean of 120 and 111 resulting in about 116.

Another idea is to vary the confidence level p when calculating the confidence intervals and searching for a value p where upper bound of the first and lower bound of the second meet. Again, we get about 116 in this specific case.

Question: What is the correct way to do this? I wasn't even sure what keywords I should use with Google.

## Edit:

I tried the suggestion of Jan Sila, minimizing the area of each pdf on the wrong side of the separation line. This seems to give almost exactly the point where both pdfs cross each other.

## Apendix:

Here is the Mathematica code:

d = MixtureDistribution[{0.8, 0.2}, {NormalDistribution[100, 10],
NormalDistribution[150, 20]}];

col = ColorData[97];
Plot[Evaluate[
{Flatten@List[#1*Function[d, PDF[d, x]] /@ #2] & @@ d, PDF[d, x]}],
{x, 60, 200}, PlotRange -> All,
PlotStyle -> {col[1], col[2], Directive[Dashing[.03], Black]}
]

• So you want a value that separates the two distributions the best? So that to the left of it, there is as little as possible of yellow and to the right of the blue distribution? That, I believe, is called Gaussian separation. I found a discussion and paper from it here – Jan Sila Dec 3 '16 at 10:36
• @JanSila I tried your approach an the value that is found looks really good. – halirutan Dec 3 '16 at 12:49
• So you happy with that? Shall i perhaps put it as answer? – Jan Sila Dec 3 '16 at 15:56
• @JanSila Yes, I think I'm going to use that and if you write up an answer, I'm happy to upvote and accept it. – halirutan Dec 4 '16 at 4:11