Why Sums And Averages Tend To Look Gaussian

We study the distribution of sums and averages of independent random variables. We show that for both discrete and continuous variables the distribution is obtained via convolution of the corresponding probability mass functions or probability density functions. These repeated convolutions smooth the pmfs and pds making them Gaussian-like, as we illustrate through two examples. In addition, we show that the convolution of two Gaussians is Gaussian (so sums of independent Gaussians are also Gaussian).

[matjazmuc-]

Thanks for the great explanation! I think there is a small mistake at 28:24 -1/2 * (a^2 + (s-a)^2) should equal -a^2 + as - s^2/2.