# What can be said about the overlap of two probability distributions

Suppose I have information about the following two normal distributions:

Red   { Mean = 7.4, Std = .501 }
Green { Mean = 9.7, Std = .465 }


What can be stated about the overlapping area depicted in the diagram below:

The area of overlap is 9%.

The question I am trying to answer is whether the likelihood that the true value of the Green in the population is greater than the true value of the Red in the population.

I'd like to be able to make a claim along the lines of: There is at least a 91% likelihood that Green is greater than Red in the population. Is that claim accurate?

What is the statistical meaning or explanation of this overlap in this context?

Edit: I simplified the question and removed extraneous information. Hopefully its clearer.

• In what sense are there even "curves" here? Since each sample yields an integral quantity of balls, there can be exactly $\ldots,5, 6, 7, 8, 9, \ldots$ balls of any given color in any sample, but nothing in between. – whuber Oct 25 '16 at 22:50
• The means are calculated from the 5 samples of 200 balls. Its sort of a non-real world example, but I'm basically trying to establish that there is some population that is represented by a probability distribution. – Mark Johnson Oct 25 '16 at 22:55
• I still cannot figure out what your curves are attempting to represent. Could you clarify? – whuber Oct 25 '16 at 22:58
• The red curve is the Red ball { Mean = 7.4, Std = .501 }, Green ball { Mean = 9.7, Std = .465 }. The curves are just the standard pdfs with those parameters. General ask, is the meaning of the overlap. Maybe just totally ignore the ball example. – Mark Johnson Oct 25 '16 at 23:02
• You say "Maybe just totally ignore the ball example", but this example constitutes 90% of the current question. Is there some applied problem you are facing, which the ball example* was intended to be a "toy model" of? In any case, you will need to edit the question to remove irrelevant content and then try to clarify the underlying question. (*Note this is a well studied example.) – GeoMatt22 Oct 25 '16 at 23:13

In order for the overlap to have meaning, it would have to be relevant to some hypothesis or question about data.

To do that let us assume a context to make an example. Assuming that a Monte Carlo simulation using the displayed means and standard deviations and normal distributions were used to make a histogram, we would be generating a mixture distribution, if we did not sort by color. In that case, the overlapping tail areas would add in those histogram categories, and modern methods that find distributions would have little trouble segregating that mixture into two normal distribution models to recover the input values.

That would not tell us which color was which, but it would correctly identify that two normal distributions produced the histogram. Similarly, if we wanted to do t-testing, we could, and that would identify that the populations have different mean values, and if we did Levene's test for differences of variance, we could then do that test.

As it stands, the overlap you show means little. A mixture distribution more naturally has different sample sizes in each sub-population of that mixture, i.e., $\mathrm{pdf}_{mixture}=p N(\mu_1,\sigma_1^2)+(1-p)N(\mu_2,\sigma_2^2)$, where $0<p<1$. Assuming a 1/3 Red and 2/3 Green mixture this would look like

For which the relative likelihood of the Red to Green probability components on a log plot would look like:

• I lightly edited (formatting), please revert if this changed any of your intent. I am not sure what the OP's underlying question is, but your two guesses are the same as mine (t-test / mixture pdf). I think it is worth emphasizing in your answer that for either of these cases the OP's provided parameters are insufficient: The "sample sizes" (and/or mixture fractions) would also be needed. – GeoMatt22 Oct 26 '16 at 17:42
• @GeoMatt22 Go ahead and make that edit. If I do it, your credits for editing will be erased. We have a paper to write, if you are willing, how do I contact you? BTW, I am the communicating author of journals.plos.org/plosone/article?id=10.1371/… if you wish to contact me. – Carl Oct 26 '16 at 18:09
• You answered my underlying question, even if it doesn't make sense to anybody else. The overlapping area is relevant/useful when you can use it to as a clue in the context of a bigger analysis. The mixture distribution is a good example. Thanks. Good luck with the paper. :) Glad my random question sparked something useful. – Mark Johnson Oct 27 '16 at 1:57
• @MarkJohnson Actually, the potential paper is about something else, not totally unrelated. BTW, +1 on your question, your initial hypothesis needed a bit of cleanup, but fortunately GeoMatt22 and I could see through haze. Welcome to the site, it is a great place to learn stats, and to have your concepts honed by having your feet put to the fire. Actually, using statistical language correctly is not the first think one learns by self-study, so the criticisms one gets here are priceless for hoof-in-mouth extraction. – Carl Oct 27 '16 at 16:52