Given a random variable $X$ with cdf that is not smooth, I read that we can construct a random variable denoted as $X\cdot M$ with smooth cdf and smooth inverse cdf, where $M$ is a random variable supported in $[0,1]$ having smooth cdf. The reference refers to this as a standard mollifier argument and does not provide further details. My question is how this construction works. Is the $\cdot$ merely a product or denotes something else? How does one show the claim? Thank you for any suggestions or reference provided.
2026-03-28 15:18:56.1774711136
Mollified random variables
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