Obviously, the CLT works for most distributions. However, I need help coming up with a non-standard (ex, Poisson, Chi-Square, Gamma, etc.) PDF that "beats" the CLT for moderately sized samples ($n=10$, $n=20$, and if possible $n=30$).
I have tried $p(x)=c \cdot x^2$ $p(x)=c\cdot x^11$, $p(x)=c\cdot e^x$ all where $0 \leq x\leq 2$.
If someone could please point me in the right direction I would really appreciate it. (Preferably a distribution that is easy to integrate, as integration isn't my strongest point)