Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
| (i) | $X, Y : \Omega \to \mathbb{R}$ are each a D3161: Random real number on $P$ |
| (ii) | $X, Y$ is an D2713: Independent random collection on $P$ |
| (iii) | $f_X$ is a D209: Probability density function for $X$ |
| (iv) | $F_Y$ is a D205: Probability distribution function for $Y$ |
Then $x \mapsto \int_{\mathbb{R}} f_X(x - y) \, d F_Y(y)$ is a D209: Probability density function for $X + Y$.