Let $x$ be a constant in $\mathbb{R}$.

Let $y$ be a random number that is generated according to a certain probability distribution.

I want to make the sum

$$x + y$$

However, I am not sure how to make it clear that $y$ is a random number (without using the English sentence that tells the reader that it is a random number).

I tried to write

$$x + y, y \sim f$$

where $f$ is the PDF or the PMF of the random variable associated with $y$.

However, here $y$ is the realization of the underlying random variable, not the random variable itself, so writing $y \sim f$ is abuse of notation.

Should I write something like

$$x + y, Y \sim f(y)$$

instead? However, now I have to make clear the connection between $Y$ (the RV) and $y$ (its realization), and how would I go about doing that using a concise mathematical notation?

Can anyone help?

  • 1
    $\begingroup$ If $y$ is a particular value of a random variable, it is no longer random, so you can write $x+y$. On the other hand, if $y$ is meant to be the random variable itself, I would write $x + Y$. $\endgroup$ – Jair Taylor Mar 24 at 4:39
  • $\begingroup$ Why in the world would you want to avoid writing a clear English sentence that defines your system of interest? $\endgroup$ – Michael Mar 24 at 5:16
  • $\begingroup$ Recall that a random variable $Y:S\rightarrow \mathbb{R}$ is a measurable function defined over the outcome space $S$. If you want a realization associated with a particular outcome $\omega \in S$ you can use $Y(\omega)$. Or use $\omega^*$ and $Y(\omega^*)$ if you like. $\endgroup$ – Michael Mar 24 at 5:21
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    $\begingroup$ @Michael Be nice. Learning to write math well is not easy. I find that many students, including my past self, go through the steps 1) describe math very imprecisely in words, receive poor marks 2) come to believe that math must be done symbolically to be rigorous 3) eventually improve their technical writing skill, where notation is a way of expressing complex thoughts in a compact way that is a supplement to, and not a replacement for, ordinary language that is sufficiently clear and precise. $\endgroup$ – Jair Taylor Mar 24 at 21:09
  • $\begingroup$ @JairTaylor : Thanks for your note. Your steps describe a process and they show more clearly my sentiments that clear and precise ordinary language is indeed part of math. $\endgroup$ – Michael Mar 24 at 22:12

In the context of probability, it is customary to use capitalized, italicized letters - generally X and Y - to denote random variables.

Discrete Random Variables


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