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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)

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If you need a distribution with finite variance for which the sum of an i.i.d. sample of size $30$ is not approximately normal, then $\operatorname{Bernoulli}(1/30)$ will serve. In general, look at $\operatorname{Bernoulli}(1/n).$ The distribution of the sum of an i.i.d. sample of size $n$ is approximately $\operatorname{Poisson}(1).$

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  • $\begingroup$ Edits have been made to the original question $\endgroup$ – Amanda R. Mar 28 '17 at 4:47

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