Convolution of probability distributions

Definition

Let $\mu$ and $\nu$ be probability distributions 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, 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 $\{2, \dots, 12\}$.