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I got introduced to the characterization of probability distributions as follows: If a random variable $X$ is given, we determine the behaviour of its probability distribution and then write e.g. $X\sim\text{Uniform}$ or $X\sim\text{Bernoulli}(n,p)$.

The exact definitions are rather descriptive: For a random variable $X$ with values in $\mathbb{R}$ it is $X\sim\textbf{Uniform}(a,b)$ if its density function $f_X$ is given by $$f_X(x)=\frac{1}{b-a}\chi_{\left[a,b\right]}(x).$$ If $X$ is discrete however, another definition is given, but only if $X$ has values in a finite set.

So, I wondered: The definitions always depend on a random variable $X$ (at least its range). However, we then write $X\sim \dots$$, implying the existence of an equivalence relation and thereby non-dependence of the exact situation. Is there a mathematical object for each important class of distributions that we can check other distributions for equivalence against?

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    I am not sure whether I understand your question. However, the ~ part is more a notation than an equivalent relationship, at least in introductory statistics textbook. Furthermore, by definition, the range is a random variable is contained in $\overline{R}$. Lastly, uniform distribution is a little bit peculiar as it can refers to a continuous uniform distribution and a discrete uniform distribution. – 2017-02-20
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    The cumulative distribution function and the characteristic function each uniquely define univariate distributions (apart from on sets of probability $0$) – 2017-02-20
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    Thanks, both. I got how distributions are uniquely defined by the density function and the cumulative distribution. My question is rather: Is there a formal object that could be called "Uniform" that all and only all uniform distributions relate to? I wonder because in our course we introduced uniform distributions multiple times, once for each domain of the distributions we worked with. – 2017-02-21

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