Timeline for What does it mean for a probability distribution to not have a density function?
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| Aug 19, 2022 at 13:45 | comment | added | whuber♦ | Continuous is not quite the opposite of "discrete." Lebesgue's Decomposition Theorem shows any measure is a mixture of three types: discrete, absolutely continuous, and "singular." The sense of "continuous" is clearly implied by the context. | |
| Aug 19, 2022 at 9:04 | comment | added | jochen | @whuber ah yes, I used "continuous" in the sense described at en.wikipedia.org/wiki/Continuous_or_discrete_variable , i.e. the opposite of "discrete". It is not clear to me which sense of "continuous" the OP intends. My second example is inspired by the sentence "... situation where we are dealing with distributions supported by low dimensional manifolds ..." in the introduction of the paper the OP links to. | |
| Aug 18, 2022 at 16:24 | comment | added | whuber♦ | Thank you for your thoughtful contributions. However, the senses in which you use "continuous" are not that of the "continuous random variable" in the question. A continuous random variable is one with a continuous distribution function, which is not the case with either of your examples. The first is a mixture of an absolutely continuous and discrete distribution while the second is a singular bivariate distribution. | |
| Aug 18, 2022 at 12:16 | history | answered | jochen | CC BY-SA 4.0 |