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Suppose we have a random variable $X$ with probability density function $f(X)$. Let $Y$ be the sum of $N$ draws of $X$. What is the PDF of $Y$?

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Well it depends. If your draws $x_1, x_2, \dots, x_n$ are taken independently then the sum $y_n=\sum_{k=1}^nx_k$ follows a law with density function $g_n$ such that

$$g_1 =f$$ $$g_n = g_{n-1}*f$$ where $*$ is the convolution operator. So this mean that the PDF $g_n$ is given recursively by:

$$g_1(x)=f(x)$$ $$ g_2(x) = (f*f)(x)=\int_\mathbb{R}f(x-u)f(u)du $$ $$ g_n(x) = (g_{n-1}*f)(x)=\int_\mathbb{R}g_{n-1}(x-u)f(u)du $$ You can check this post as well:

How do you sum PDF's of random variables?

If your draws are not taken independently, you cannot say anything a priori about the law of the sum.

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  • $\begingroup$ Thanks Maxime! I was thinking vaguely Central Limit Theorem but your answer is much more what I was hoping for. $\endgroup$ – Lars Ericson Oct 27 '17 at 17:13
  • $\begingroup$ Also Maxime note that this answer was helpful to me for a more constructive/computation application of this topic here.. $\endgroup$ – Lars Ericson Oct 28 '17 at 14:15

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