What does $U(y)$ mean in an equation

I am reading a book on probability (Probability, Random Variables and Stochastic Processes by A. Papoulis) and lets just say that notation is not its strong suit.

Lately this symbol has been popping up. I know that $$X \sim U(a,b)$$ means that the random variable $$X$$ is uniformly distributed at $$(a,b)$$, but I can't understand what it means in an equation.

e.g (from a problem in the book)

$$f_{Y}(y) = e^{-y}U(y)$$

• In modern probability textbooks, random variables are usually capitalised, whereas small letters are rather used for realisations of these random variables. Apr 29 '19 at 12:45
• I know but I do not think that this is applicable here since there is no random variable u Apr 29 '19 at 12:46
• You've written "random variable x". Apr 29 '19 at 12:48
• Oh I see what you meant. Edited the question Apr 29 '19 at 12:50
• Please use MathJax for math expressions so that your question is correctly parsed by the site. Btw, you may consider using blockquotes > in Markdown for quoting external contents. Apr 29 '19 at 12:59

Without having the text at hand:

1. It could be that they mean the density function of a random variable which is distributed according to $$U(a,b)$$.

2. Depending on context it could be a different function $$U$$, but I'd doubt that.

Are the two $$U$$'s written the same way? Sometimes the "distribution-naming" letters are written with more fancy, e.g. $$\mathcal{U}$$ (or similar).

• Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on MathJax notation, MathJax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Apr 29 '19 at 12:45
• I do not see how U(y) defines a uniform distribution. What is the interval if that is the case? Apr 29 '19 at 12:49
• That would be dependent on the context. If I was speaking about a random variable $x \sim U(a,b)$ and in the next line there appeared the function you were talking about, I would assume we are still talking about $U(a,b)$ (with $a,b$ either as general variables or concrete values). Apr 29 '19 at 12:53
• In the problem $X$ is a random variable with $X \sim U(0,1)$ and $Y = -ln X$ is another random variable. $f_{Y}(y)$ is the density of $Y$ Apr 29 '19 at 12:58
• I have tried to improve the readability of your question by introducing $\rm \LaTeX$. It is possible that I unintentionally changed the meaning of your question. Please proofread the question to ensure this has not happened. Please use math markup for each math expression to ensure the correct parsing of your content. Apr 29 '19 at 13:05