# Defining the overlapping area of two log-normal distributions with different means, same variance, and different scaling factors that add up to 1

Define $$\begin{cases} X_1\sim Lognormal(ln(\mu_1), \sigma^2) \\ X_2\sim Lognormal(ln(\mu_2), \sigma^2) \end{cases}$$ where $$\mu_2>\mu_1>0$$ and that there is a definite proportion, $$\eta\in(0,1)$$, between $$X_1$$ and $$X_2$$ such that $$\begin{cases} f_1(x)=\frac{\eta}{x\sigma\sqrt{2\pi}}e^{-{\frac{(ln(x)-ln(\mu_1))^2 \,\,\,\,\,\,}{2\sigma^2}}} \\ f_2(x)=\frac{1-\eta}{x\sigma\sqrt{2\pi}}e^{-\frac{(ln(x)-ln(\mu_2))^2 \,\,\,\,\,\,}{2\sigma^2}} \end{cases}$$ where $$f_1$$ and $$f_2$$ represent the $$\eta$$-scaled PDF's of $$X_1$$ and $$X_2$$, respectively.

Based on the above definitions, note that $$\int_{x=0}^\infty f_1(x)\,dx\,+\int_{x=0}^\infty f_2(x)\,dx=1$$.

Given $$\mu_1$$, $$\mu_2$$, $$\sigma$$, and $$\eta$$, how is the overlapping area of the two probability distribution curves, $$OVL=f(\mu_1,\mu_2,\sigma,\eta)$$, defined?

Please see a illustrative plot below, where $$OVL=f(\mu_1=5,\mu_2=10,\sigma=20\%,\eta=50\%)$$ is highlighted in yellow: I am able to perform numerical approximation for $$OVL$$ using the trapezoidal rule, but I need to express $$OVL$$ explicitly and I am not sure how to do so.

• – user2974951 Oct 15 '20 at 6:03
• Thanks. I have seen these, but they refer to normal distributions and also do not have the proportion parameter defined in my question. – Matthew Hui Oct 15 '20 at 7:01
• What statistical meaning do you suppose the yellow area to have? (I cannot recognize any -- and it is not equivalent to any function of your $f_1$ and $f_2.$) What is the function $f$ you want to integrate?? – whuber Oct 15 '20 at 16:12
• Thank you for the references. The problem is that OVL does not transform meaningfully when you analyze the distribution of $\exp(X)$ (the lognormal distribution) rather than $X$ itself (the normal distribution), so we're stuck right at the beginning: what do you hope this "overlapping area" represents? – whuber Oct 16 '20 at 14:00

With help from a friend we solved the problem ourselves:

Let $$\{\tau\in\Bbb{R}^+|f_1(\tau,\mu_1,\sigma,\eta)=f_2(\tau,\mu_2,\sigma,\eta)\}$$, then \begin{align} \frac{\eta}{\tau\sigma\sqrt{2\pi}}e^{-\frac{(\ln\tau-\ln{\mu_1})^2}{2\sigma^2}}&=\frac{1-\eta}{\tau\sigma\sqrt{2\pi}}e^{-\frac{(\ln\tau-\ln{\mu_2})^2}{2\sigma^2}} \\ e^\frac{(\ln\tau-\ln{\mu_1})^2-(\ln\tau-\ln{\mu_2})^2}{2\sigma^2}&=\frac{\eta}{1-\eta}\quad (\because\tau>0) \\ \frac{2\ln\tau\ln{\tau_2}-2\ln\tau\ln{\tau_1}+(\ln{\mu_1})^2-(\ln{\mu_2})^2}{2\sigma^2}&=\ln\frac{\eta}{1-\eta} \\ \ln\tau&=\frac{\sigma^2[\ln\eta-\ln(1-\eta)]}{\ln{\mu_2}-\ln{\mu_1}}+\frac{\ln{\mu_2}+\ln{\mu_1}}{2} \\ \tau&=e^{\frac{\sigma^2[\ln\eta-\ln(1-\eta)]}{\ln{\mu_2}-\ln{\mu_1}}+\frac{\ln{\mu_2}+\ln{\mu_1}}{2}} \end{align} Since $$\mu_2>\mu_1$$ and $$\eta\in(0,1)$$, $$\exists!\tau\in\Bbb{R}^+|f_1=f_2$$

$$\forall x\in\Bbb{R}^+$$,

$$\ln{f_2(x,\mu_2,\sigma,\eta)}-\ln{f_1(x,\mu_1,\sigma,\eta)}$$

$$=\ln\frac{1-\eta}{x\sigma\sqrt{2\pi}}-\ln\frac{\eta}{x\sigma\sqrt{2\pi}}+\frac{(\ln{x}-\ln{\mu_1})^2}{2\sigma^2}-\frac{(\ln{x}-\ln{\mu_2})^2}{2\sigma^2}$$

$$=\ln\frac{1-\eta}{\eta}+\frac{(\ln{\mu_1})^2-(\ln{\mu_2})^2}{2\sigma^2}+\frac{\ln{\mu_2}-\ln{\mu_1}}{\sigma^2}\ln{x}$$, which is strictly increasing on $$x\quad(\because\mu_2>\mu_1)$$

$$\therefore\frac{f_2(x,\mu_2,\sigma,\eta)}{f_1(x,\mu_1,\sigma,\eta)}$$ is strictly increasing on $$x$$

It follows that:

$$\begin{cases} f_2(x,\mu_2,\sigma,\eta)>f_1(x,\mu_1,\sigma,\eta)\quad\forall x>\tau \\ f_2(x,\mu_2,\sigma,\eta)

The $$\eta$$-scaled CDF's of $$X_1$$ and $$X_2$$, denoted as $$F_1$$ and $$F_2$$, are defined as follows: $$\begin{cases} F_1(x,\mu_1,\sigma,\eta)=\frac{\eta}{2}[1+erf(\frac{\ln{x}-\ln{\mu_1}}{\sqrt{2}\sigma})] \\ F_2(x,\mu_2,\sigma,\eta)=\frac{1-\eta}{2}[1+erf(\frac{\ln{x}-\ln{\mu_2}}{\sqrt{2}\sigma})] \end{cases}$$, where $$erf(z)=\frac{2}{\sqrt\pi}\int_0^z{e^{-t^2}dt}$$

By (*),

\begin{align} OVL=f(\mu_1,\mu_2,\sigma,\eta)&=F_2(\tau,\mu_2,\sigma,\eta)+[1-F_1(\tau,\mu_1,\sigma,\eta)] \\ &=1+F_2(\tau,\mu_2,\sigma,\eta)-F_1(\tau,\mu_1,\sigma,\eta) \end{align}