Symmetry of the Distribution of Sample Mean

Question

I've studied all the proofs and the theorems. The central limit theorem and the laws of mean and variance, but i still don't grasp intuitively why the distribution of Sample Mean tends to a symmetrical distribution ?

Are you able to help me ? Thanks

Answer 1

The pdf of the sum of two random variables is the result of the convolution of the two initial pdf's, which can be seen as a low-pass filtering of the first distribution with the second, mirrored (in other terms, a sliding weighted average). The mirroring comes from the fact that you accumulate for constant sums ($s=x+y\implies y=s-x$).

The result is a smoother and more symmetric function. Smoother because you are averaging with positive-only coefficients, and more symmetric because of the mirroring (high functions values get a lower weight, and conversely).

The limit pdf is a Gaussian because this function enjoys a special property: the result of a Gaussian convolved with a Gaussian is another Gaussian.