A pretty reason why Gaussian + Gaussian = Gaussian
The Central Limit Theorem posits that the sum of independent random variables converges to a Gaussian distribution under appropriate conditions due to rotational symmetry in the convolution integral space. This stability arises because the Gaussian function is uniquely characterized by this geometric property, making it an attractor for repeated convolutions within probability theory and statistical inference. Consequently, the specific functional form $e^{-x^2}$ emerges not as an arbitrary choice but as a necessary fixed point required to satisfy the universality claims of the theorem.
A pretty reason why Gaussian + Gaussian = Gaussian
The Central Limit Theorem posits that the sum of independent random variables converges to a Gaussian distribution under appropriate conditions due to rotational symmetry in the convolution integral …