Let $X$ be a continuous random variable which induces a probability measure on $\mathbb{R}$ denoted by $\mu$. Are there any instances in statistics when we deal with random variables $X$ such that $\frac{d\mu}{d\lambda}$ does not exist (where $\lambda$ is the usual Lebesgue measure)?

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    $\begingroup$ A continuous random variable is a random variable whose cumulative distribution function is continuous everywhere, this does not imply it is differentiable anywhere. See, e.g., the Cantor function. $\endgroup$
    – Xi'an
    Dec 10, 2022 at 10:26
  • $\begingroup$ See also stats.stackexchange.com/a/585601/7224 $\endgroup$
    – Xi'an
    Dec 10, 2022 at 10:34
  • $\begingroup$ @Xi'an That was not my question, " are there instances in statistics " , not in mathematics in general, i.e. are such distributions ever used in statistical analysis? $\endgroup$ Dec 10, 2022 at 15:01
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    $\begingroup$ I do not have an example in dimension one, but some copulas are not everywhere differentiable, see the above link. $\endgroup$
    – Xi'an
    Dec 10, 2022 at 17:27
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    $\begingroup$ @Xi'an That is a good example thank you. $\endgroup$ Dec 10, 2022 at 17:45


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