# What is the PDF of the sum of N drawings from a PDF

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 $Y$?

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: