---
tag: 07CHJ5YX28
kind: definition
name: Convolution of probability distributions
msc: ["60E05"]
defines: 0F8GDKM0A7
uses: [0QB913WM5E, 0E9502NT92, 0WF4EJP91W, 04YG690N5V]
sources:
  - url: https://en.wikipedia.org/wiki/Convolution_of_probability_distributions
    locator: Introduction
---

## Definition

Let $\mu$ and $\nu$ be [[probability distributions|0QB913WM5E]] on $\mathbb{R}$.
Their convolution is the probability distribution $\mu * \nu$ on $\mathbb{R}$ with
$$(\mu * \nu)(B) = \int_{\mathbb{R}} \nu(B - x) \, \mu(dx)$$
for each Borel set $B \subseteq \mathbb{R}$, where $B - x = \{b - x : b \in B\}$.

## Remarks

If $X \sim \mu$ and $Y \sim \nu$ are [[independent|0E9502NT92]], then $X + Y \sim \mu * \nu$.
For distributions on the integers with mass functions $p$ and $q$, $\mu * \nu$ has the mass function $(p * q)(n) = \sum_{k} p(k) \, q(n - k)$.
If $\mu$ and $\nu$ have densities $f$ and $g$, then $\mu * \nu$ has the density $(f * g)(x) = \int f(y) \, g(x - y) \, dy$.
The convolution is commutative and associative. Its characteristic function is the product: $\varphi_{\mu * \nu} = \varphi_\mu \varphi_\nu$.

## Explanations

### L1 human

The convolution answers the question: "What is the law of $X + Y$, if $X$ and $Y$ are independent?"
For two fair dice, the convolution of the two uniform laws on $\{1, \dots, 6\}$ gives the familiar triangle-shaped law of the sum on $\{2, \dots, 12\}$.