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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)

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

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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).

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