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What is the CDF and the PDF (or approximation) of the sum of two truncated gaussian $X = TN_x(\mu_x,\sigma_x;a_x,b_x)$ and $Y = TN_y(\mu_y,\sigma_y;a_y,b_y)$ ? where $TN(\mu,\sigma;a,b)$ is a truncated normal distribution where a and b are the the lower and upper bounds of the truncation respectively.

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  • $\begingroup$ Just to clarify, you want to sum distributions or want to know distribution of sum of variables? Because these are different things. Even in not truncated case sum of normal is normal but sum of normal distributions is not normal. $\endgroup$ – Alexander Vigodner Jun 12 '14 at 13:03
  • $\begingroup$ I want to know the distribution of sum of variables. Sorry for the lack of precision. $\endgroup$ – user51213 Jun 12 '14 at 13:50
  • $\begingroup$ Are they independent? Because then we can use the formula:$P_z(z)=E(P_x(z-Y))=E(P_y(z-X)$ where $Z=X+Y$, $P_{x,y,z}$ are CDFs of $x,y,z$ respectively, $E$ is the mean operator. $\endgroup$ – Alexander Vigodner Jun 12 '14 at 15:34
  • $\begingroup$ Are either $a_x$ or $a_y$ positive? $\endgroup$ – wolfies Jun 12 '14 at 15:36
  • $\begingroup$ Yes, they are independent. What is the form of $P_x(z - Y)$. $\endgroup$ – user51213 Jun 12 '14 at 16:23
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This is not an easy problem to obtain a closed-form solution to. As always, there are a number of different approaches, but unfortunately many of them seem to yield intractable outcomes. The approach taken here is to proceed manually step-by-step using the Method of Transformations, aided by using a computer algebra system to do the nitty-gritties where that is helpful.

Given: Let:

  • $X \sim N(\mu_1, \sigma_1^2)$ be doubly truncated (below and above) at $(a_1, b_1)$, where $0<a_1<b_1$, and

  • $Y \sim N(\mu_2, \sigma_2^2)$ be doubly truncated (below and above) at $(a_2, b_2)$, where $0<a_2<b_2$.

Here is an illustrative plot of the doubly truncated Normal pdf, given different parameter values:

enter image description here

Solution

By virtue of independence, the joint pdf of $(X,Y)$, say $f(x,y)$ is the product of the individual pdf's:

enter image description here

where Erf[.] denotes the error function.

Step 1: Let $Z=X+Y$ and $W=Y$. Then, using the Method of Transformations, the joint pdf of $(Z,W)$, say $g(z,w)$, is given by:

enter image description here

where:

  • Transform is a function from the mathStatica package for Mathematica, which automates the calculation of the transformation and required Jacobian.

  • the transformation $(Z=X+Y, W=Y)$ induces dependency in the domain of support between $Z$ and $W$. In particular, since $X$ is bounded by $(a_1,b_1)$, it follows that $Z=X+Y$ is bounded by $(a_1+W, b_1+W)$. This dependency is captured using the Boole statement in the line above which acts as an indicator function. Because the dependency has been captured into pdf g itself, we can enter the domain of support for pdf g in standard rectangular fashion as:

enter image description here

Step 2: Given the joint pdf $g(z,w)$ just derived, the marginal pdf of $Z$ is:

enter image description here

which yields the 4-part piecewise solution:

enter image description here

... defined subject to the domain of support: $a_1 + a_2 < Z < b_1 + b_2$.

The solution sol appears very small on screen: to view it larger, either open the image in a new window, or click here.

The Marginal computation is far from simple: it takes about 10 minutes of pure computing time to solve on the new R2-D2 Mac Pro.

All done.

Illustrative Plot

Here is a plot of the solution pdf of $Z = X + Y$, for the same parameter values used in the first diagram:

  • $(\mu_1 = 5, \sigma_1 = 2, a_1 = 5, b_1 = 8)$
  • $(\mu_2 = 1, \sigma_2 = 4, a_2 = 7, b_2 = 9)$

enter image description here

Monte Carlo Check

It is always good idea to check symbolic work with a quick Monte Carlo comparison, just to make sure that no mistakes have crept in. Here is a comparison of the empirical pdf (Monte Carlo simulation of the sum of the doubly truncated Normals) [the squiggly blue curve] plotted together with the exact theoretical pdf (sol) derived above [ red dashed curve ].

enter image description here

Looks fine!

Notes

  1. The Marginal function used above is also from the mathStatica package for Mathematica. As disclosure, I should add that I am one of the authors of the software used.
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OK, I believe the variables are independent. Then let $\Phi(x,\mu,\sigma)$ be normal CDF and $\rho(x,\mu,\sigma)$ it's density $$ \rho(x,\mu,\sigma)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$ Thus $$ P_x(x)=(x\le a_x)\Phi_(a_x,\mu_x,\sigma_x)+(a_x < x < b_x)\int_{a_x}^x \rho(x,\mu_x,\sigma_x) dx+(x>b_x)(1-\Phi(b_x,\mu_x,\sigma_x)) $$ $$ P_y(y)=(y\le a_y)\Phi_(a_y,\mu_y,\sigma_y)+(a_y < y < b_y)\int_{a_y}^y \rho(y,\mu_y,\sigma_y) dy+(y>b_y)(1-\Phi(b_y,\mu_y,\sigma_y)) $$ So the mean operator $E$ will look as follows: $$ E(f(X))=\Phi(a_x,\mu_x,\sigma_x)f(a_x)+\int_{a_x}^{b_x} f(x)\rho(x,\mu_x,\sigma_x) dx+f(b_x)(1-\Phi(b_x,\mu_x,\sigma_x)) $$ Now using the formula $P_z(z)=E(P_y(z-X))$ and defining $f(x)=P_y(z-x)$ we can write the following $$ P_z(z)=\Phi(a_x,\mu_x,\sigma_x)P_y(z-a_x)+\int_{a_x}^{b_x} P_y(z-x)\rho(x,\mu_x,\sigma_x) dx+P_y(z-b_x)(1-\Phi(b_x,\mu_x,\sigma_x)) $$ May be going through different inequalities you can simplify this expression, plugging the expression ofr $P_y$ into the last formula.

EDIT As was noted by wolfies I wrote $P_{x,y}$ for censored variables rather truncated. Thank you for catching me. We can rewrite truncated distribution even simpler: $$ P_y(y)=\frac{1}{\Phi(b_y,\mu_y,\sigma_y)-\Phi(a_y,\mu_y,\sigma_y)}\int_{\max(y,a_y)}^{\min(y,b_y)} \rho(y,\mu_y,\sigma_y) dy =(y>a_y) \frac{\Phi(\min(y,b_y),\mu_y,\sigma_y)-\Phi(a_y,\mu_y,\sigma_y)}{\Phi(b_y,\mu_y,\sigma_y)-\Phi(a_y,\mu_y,\sigma_y)} $$ and $$ E(f(X))=\frac{1}{\Phi(b_x,\mu_x,\sigma_x)-\Phi(a_x,\mu_x,\sigma_x)}\int_{a_x}^{b_x} f(x)\rho(x,\mu_x,\sigma_x) dx $$ So $$ P_z(z)=\frac{1}{\Phi(b_x,\mu_x,\sigma_x)-\Phi(a_x,\mu_x,\sigma_x)}\int_{a_x}^{b_x} P_y(z-x)\rho(x,\mu_x,\sigma_x) dx $$

And finally we have very similar formula for density $$ \rho_z(z)=\frac{\int_{a_x}^{b_x} (a_y<z-x<b_y)\rho(z-x,\mu_y,\sigma_y)\rho(x,\mu_x,\sigma_x) dx}{(\Phi(b_x,\mu_x,\sigma_x)-\Phi(a_x,\mu_x,\sigma_x))(\Phi(b_y,\mu_y,\sigma_y)-\Phi(a_y,\mu_y,\sigma_y))} $$

It can be simlified further $$ \rho_z(z)=\frac{\int_{\max(a_x,z-b_y)}^{\min(b_x,z-a_y)} \rho(z-x,\mu_y,\sigma_y)\rho(x,\mu_x,\sigma_x) dx}{(\Phi(b_x,\mu_x,\sigma_x)-\Phi(a_x,\mu_x,\sigma_x))(\Phi(b_y,\mu_y,\sigma_y)-\Phi(a_y,\mu_y,\sigma_y))} $$ The last result almost surely can be completed in $\Phi$ terms.

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  • $\begingroup$ The cdf of the doubly truncated Normal, evaluated at the lower bound $X = a_x$ is $P_x(a_x) = 0$, ... By contrast, what you have described in your cdf $P_x(x)$ is the cdf of a doubly censored Normal ... not a doubly truncated Normal. $\endgroup$ – wolfies Jun 12 '14 at 16:51
  • $\begingroup$ Opps.You are right. OK let me edit my answer then $\endgroup$ – Alexander Vigodner Jun 12 '14 at 16:56
  • $\begingroup$ Thank you. Could you help me calculate $P_z(z)$ ? Thanks in advance. $\endgroup$ – user51213 Jun 13 '14 at 14:44
  • $\begingroup$ Well, I gave finally formula for the density. These are pure exponents of $-(z-x)^2,-x^2)$. They can be recombined to get the final result in terms of $\Phi$ . $\endgroup$ – Alexander Vigodner Jun 13 '14 at 19:53

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