# Singular Distributions in Statistics

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)?

• 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. Commented Dec 10, 2022 at 10:26
• Commented Dec 10, 2022 at 10:34
• @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? Commented Dec 10, 2022 at 15:01
• I do not have an example in dimension one, but some copulas are not everywhere differentiable, see the above link. Commented Dec 10, 2022 at 17:27
• @Xi'an That is a good example thank you. Commented Dec 10, 2022 at 17:45

A simple example is the uniform distribution on the line $$y=x$$ for $$0\le x, y \le 1$$, which has cdf $$F(x,y)=\min(x,y)$$. For other examples see What does it mean for a probability distribution to not have a density function? and references there.