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This is one interesting question I take some time to search if there is any distribution function that is continuous but without the PDF.

After some search I found Cantor distribution, sometimes referred as the Devil's staircase according to Wikipedia.

The reason why I am asking this question are some comments I got from one member of this website saying there are many of these.

There are some distributions like lambda distribution with if statement inside the PDF which may be what he was referring.

I am also not sure if he assumed some conditional distributions in which case it may be the PDF is possible unknown.

Easy Points
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    See [this](https://stats.stackexchange.com/questions/455668/defining-continuous-random-variables-via-uncountable-sets) and its links. – kjetil b halvorsen Mar 02 '21 at 01:39
  • I like your link and specifically a part where it reads "Such singular distributions are not common in statistics". Thank you @kjetilbhalvorsen – Easy Points Mar 02 '21 at 02:00
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    So as not to cherry-pick, I'm quoting the full sentence here: "Such singular distributions are not common in statistics (except as counterexample), but are ubiquitous in other areas." – Arya McCarthy Mar 02 '21 at 02:24
  • Also see @kjetilbhalvorsen's comment at https://stats.stackexchange.com/questions/298293/absolutely-continuous-random-variable-vs-continuous-random-variable/298434#comment567156_298434. – whuber Mar 02 '21 at 14:58
  • @whuber to me, this link doesn't answer this question, I do not see in there concrete link to any particular distribution without PDF. I hoped someone will answer what does it mean that the distribution doesn't have the PDF. I have the strong pre-conviction that statisticians really need continuous distributions with PDF so that they can express and calculate `d*`, `p*`, `q*` functions in R for instance. Without that this would not be possible. – Easy Points Mar 08 '21 at 00:34
  • Your question doesn't ask for an example; but if you want one, see https://stats.stackexchange.com/questions/103969. – whuber Mar 08 '21 at 14:37
  • This was an example to the discrete RV. In my understanding discrete variables do not have PDF. I don't think PDF is the same as PMF, but I don't know much on that topic. I may be wrong. – Easy Points Mar 11 '21 at 00:18

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