[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

The support of a function [math]f(x)[/math] is defined to be the set

[[math]] \{x\ :\ f(x) \gt 0\}\ . [[/math]]

Suppose that [math]X[/math] and [math]Y[/math] are two continuous random variables with density functions [math]f_X(x)[/math] and [math]f_Y(y)[/math], respectively, and suppose that the supports of these density functions are the intervals [math][a, b][/math] and [math][c, d][/math], respectively. Find the support of the density function of the random variable [math]X+Y[/math].