Hey there, let's unpack this question clearly—since you're diving into probability integral transforms and copulas, getting this notation right will help keep your reasoning tight.

First, let's clarify the literal difference between the two notations:

• U(0,1) refers to a uniform distribution over the open interval (0,1), meaning the random variable can take any value strictly between 0 and 1, but never exactly 0 or 1.
• U[0,1] refers to a uniform distribution over the closed interval [0,1], so the random variable can take every value from 0 up to and including 1, including the two endpoints.

Now, why do we often use U[0,1] specifically in the context of the probability integral transform? Let's tie this to the distribution function $F$ you mentioned:
Any valid distribution function $F(x) = P(X \leq x)$ has a range of exactly $[0,1]$. It can hit 0 (when $x$ is smaller than all possible values of $X$) and 1 (when $x$ is larger than all possible values of $X$), and it's right-continuous, so those endpoint values are attainable.

The probability integral transform works by mapping a random variable $Y$ to $F_Y(Y)$, which produces a uniform random variable. Since $F_Y(Y)$ can actually equal 0 or 1 (in cases where $Y$ hits its minimum or maximum possible value), using U[0,1] is more mathematically consistent—it matches the exact range of the transformed value.

That said, for continuous random variables, the probability of hitting exactly 0 or 1 is 0, so in practice, the two notations behave almost identically for calculations. But using U[0,1] avoids a logical inconsistency: if we claimed the transformed variable follows U(0,1), we'd be saying it can never take values that the transform does produce (even if those cases have zero probability).

To tie this to the example you shared:

Realize that the range of any distribution function $F$ is $[0,1]$, which equals the domain of any random variable governed by $\text{Uniform}$(0,1).

You might notice a small mismatch here—the range is a closed interval, but the notation uses an open interval. This is likely a minor notation slip; the intended domain should be the closed interval $[0,1]$ to perfectly align with the range of $F$, which is exactly why most rigorous treatments prefer U[0,1] for this context.

备注：内容来源于stack exchange，提问作者Dr. Statistics