# What is the intuitive meaning behind a random variable being defined as a “lattice”?

In probability theory, a nonnegative random variable $X$ is called a lattice if there exists $d \geq 0$ such that $\sum_{n=0}^{\infty}P(X=nd) = 1$.

Is there a geometric interpretation for why this definition is called a lattice?

It means that $X$ is discrete, and there is some kind of regular spacing to its distribution; that is, the probability mass is concentrated on a finite/countable set of points ${d, 2d, 3d, \dots}$.

Note that not all discrete distributions are lattices. Eg if $X$ can take on the values $\{1, e, \pi, 5\}$, this is not a lattice since there is no $d$ such that all the values can be expressed as multiples of $d$.