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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!

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  • $\begingroup$ Yes, $Y$ is a random length vector, nothing’s wrong with that! $\endgroup$
    – Xi'an
    Feb 8 '15 at 19:51
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Your notation is perfectly fine; there is nothing wrong with specifying a random vector length. Another trick you might consider in this case, to simplify your notation, is to define an infinite sequence of uniform values, and then specify your finite random vector as you have described:

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 $x \in \mathbb{N}$).

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