# Notation for a random vector whose length depends on another random variable?

I have the following process that I'm trying to describe with random variables.

1. First, I have a random variable $X$ that takes on values drawn from a Poisson distribution with parameter $\lambda=20$. Let $X=x$ be the realized value.
2. I then sample $x$ values from a discrete uniform distribution (say, integers between 1 and 10). Let $Y$ be a possible sequence of sampled values.

What is the proper mathematical notation for describing $Y$?

If a fixed number of values $n$ were sampled, I could say that $Y=(Y_1, \ldots, Y_n)$ is a multivariate random variable of length $n$, where $\forall i \in \{1, \ldots, n\}, Y_i \sim \text{unif}\{1, 10\}$.

When $n$ is unknown, does this description make sense? $Y=(Y_1, \ldots, Y_X)$ is a multivariate random variable of length $X$, where $X \sim \text{Poisson}(20)$ and $\forall i \in \{1, \ldots, X\}, Y_i \sim \text{unif}\{1, 10\}$.

That seems correct to me, but I'm no expert, so I wanted to verify that what I wrote above is standard and correct notation. Thanks!

• Yes, $Y$ is a random length vector, nothing’s wrong with that! Feb 8, 2015 at 19:51

Consider a sequence $Y_1, Y_2, Y_3, \cdots \sim \text{IID U}\{ 1, ..., 10\}$ composed of independent uniform values, and let $X \sim \text{Pois}(\lambda)$ be an independent Poisson random variable. We will consider the random vector $\boldsymbol{Y}_X = (Y_1, Y_2, ..., Y_X)$ which has random length.
Doing it this way allows you to change $x$ later on without having to redefine the sequence values (since they are pre-defined large enough for any $x \in \mathbb{N}$).