# Calculate probability (area) under the overlapping area of two normal distributions

I have two normal distributions defined by their averages and standard deviations.

Sample 1: Mean=5.28; SD=0.91

Sample 2: Mean=8.45; SD=1.36

You can see how they look like in the next image:

How can I get the probability to obtain an individual from the overlapping area (green)? Is the probability the same as the area?

• What do you mean by the probability of obtaining an individual from the area? – Juho Kokkala Jun 18 '14 at 8:37
• If you sample points from either normal distribution, you get points on the Perikymata-axis rather than on the 2-dimensional area. Furthermore, the green zone is infinitely wide, so all values sampled from either distribution are under the green zone, so in this sense the probability would be 1. – Juho Kokkala Jun 18 '14 at 8:47
• @JuhoKokkala OP likely means integral over the green area. – Vladislavs Dovgalecs Nov 21 '17 at 18:40
• @VladislavsDovgalecs What would the question "Is the probability the same as the area" then mean? – Juho Kokkala Nov 21 '17 at 19:21
• @VladislavsDovgalecs I think you misunderstood my comment, I was not asking for an interpretation of the area but trying to parse the question. In any case, please note OP's comments to the accepted answer and a followup question posted here stats.stackexchange.com/questions/103821. – Juho Kokkala Nov 23 '17 at 6:14

It is not quite clear what you mean by probability to obtain an individual from the overlapping area. This solves for the area of the green zone in your diagram:

Let:

• $X_1 \sim N(\mu_1,\sigma_1^2)$ with pdf $f_1(x_1)$ and cdf $F_1(x_1)$ and

• $X_2 \sim N(\mu_2,\sigma_2^2)$ with pdf $f_2(x_2)$ and cdf $F_2(x_2)$,

where $\mu_1 < \mu_2$. In your example, the 'black variable' corresponds to $X_1$.

Let $c$ denote the point of intersection where the pdf's meet in the green zone of your plot Then, the area of your green intersection zone is simply:

$$P(X_1>c) + P(X_2<c) = 1 - F_1(c) + F_2(c) = 1-\frac{1}{2} \text{erf}\left(\frac{c-\mu _1}{\sqrt{2} \sigma _1}\right)+\frac{1}{2} \text{erf}\left(\frac{c-\mu _2}{\sqrt{2} \sigma _2}\right)$$

where erf(.) is the error function.

Point $c$ is the solution to $f_1(x) = f_2(x)$ within the green zone, which yields:

$$c = \frac{\mu _2 \sigma _1^2-\sigma _2 \left(\mu _1 \sigma _2+\sigma _1 \sqrt{\left(\mu _1-\mu _2\right){}^2+2 \left(\sigma _1^2-\sigma _2^2\right) \log \left(\frac{\sigma _1}{\sigma _2}\right)}\right)}{\sigma _1^2-\sigma _2^2}$$

For your example, with ${\mu_1 = 5.28, \mu_2 = 8.45, \sigma_1 = 0.91, \sigma_2 = 1.36}$, this yields: $c = 6.70458...$,

and the area of the green section is: 0.158413 ...

• @antecessor Depends on what you mean by 'overlapping among both'. In the question you talk about some probability, but it is unclear what probabilistic interpretation the area computed here would have. – Juho Kokkala Jun 18 '14 at 8:57
• @JuhoKokkala Ok, I will write a new question to address the exact question. – antecessor Jun 18 '14 at 10:38
• the resolution of the equation pdf1=pdf2 yields to two solutions, so we have two intersection points, why you delete the other point @wolfies – user61828 Dec 1 '14 at 11:51
• Because, within the green zone, there is only one solution. – wolfies Dec 1 '14 at 12:32
• Is the equation for c given at the bottom of the answer correct? If both distributions have equal variance, the equation requires dividing by zero! – sammosummo Mar 21 '18 at 21:28

@abdelbasset, To improve @wolfies 's answer above, there are two intersection points c, let's call them c1 and c2. Here, c1=-1.2848 and c2=6.7046. @wolfies ignored c1 (it likely will be too far in the tails of the probability density functions to matter (especially if we round to just a few decimal points)).

The more correct way is to find both c1 and c2, and to find the area of overlap of both functions:

AREA OF OVERLAP = P(X1 > j1) + P(X2 < j1) - [P(X2 < j2)-P(X1 < j2)]
therefore, using cdf's
AREA OF OVERLAP = 1 - F1(j1,μ2,σ2)+F2(j1,μ2,σ2)-F2(j2,μ2,σ2)+F1(j2,μ1,σ1)

• Welcome to the site, @luckapani. You cannot use the "Your Answer" field to comment or respond to comments. As a result, this would be deleted. However, there is the core of an answer here. Can you edit your post to make it more of an answer & not a comment? Since you're new here, you may want to take our tour, which contains information for new users. – gung - Reinstate Monica May 19 '15 at 16:12

Normalise the graphs to an area of 1 by dividing each by their respective standard deviation. Then use simple subtraction from a z-graph to calculate the probability of an occurrance in that overlap area. No need for erfs.

• i assume that he means that he wants to compute the probability in the green region. That can be done simply by adding the area in the right tail of the first distribution to the ares in the left tail of the second distribution. There is no normalization necessary. – Michael R. Chernick Nov 2 '19 at 23:36
• The separate normalization of the two graphs will change the overlap area unless they have identical SDs. Thus, this answer is generally wrong. – whuber Nov 3 '19 at 13:52