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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

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  • $\begingroup$ Why is there "symmetry" in the title and not in the body ? $\endgroup$
    – user65203
    Jul 28, 2018 at 9:42
  • $\begingroup$ Yeah, i edited thanks $\endgroup$
    – Koinos
    Jul 28, 2018 at 9:44

1 Answer 1

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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.

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