3 problems

Let $X$ and $Z$ be independent draws from arbitrary unknown probability distributions with densities $f_X$ and $f_Z$, satisfying … Let $Y$ be drawn from the mixture distribution ……

Let $x$, $y$, and $z$ be three points on the real line, where $x$ and $z$ are generated independently from two distinct distributions and $y$ is generated from their 50--50 mixture…

Powerlessness conjecture. When $\gamma<\tfrac{1}{2}$, all tests based on the vertex degrees $\{D_v(n)\}_{v\in[n]}$ are powerless, and all tests based on $G_n$ are powerless.