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There are quantitative versions of the CLT which provide an estimate of the rate of convergence. One is given at http://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem. This says that the sequence of CDFs of the rescaled sample means converges uniformly to the appropriate normal CDF, provided the third moment of the variables exists. Assuming the ...

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The joint distribution of $x$ and $y$ is circularly symmetric around the origin of the $x,y$-plane. The set of points $A$ where $x > y$ consists of all points below the line $x = y$; in polar coordinates, it is all points $(r,\theta)$ such that $r > 0$ and $-\frac34\pi < \theta < \frac14\pi.$ The probability distribution integrated over this ...

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A number of observations: (i) As functions of independent random variables, $\dfrac{X_n}{\sqrt{n}}\,\,$ is independent of $\,\,\displaystyle \sum\limits_{m=1}^{n-1}{X_m^2}\,$. (ii) $\,\,\dfrac{X_n}{\sqrt{n}}\sim\mathcal{N}\left(0, 1/n\right)\,\,$ and $\,\,\displaystyle \sum\limits_{m=1}^{n-1}{X_m^2}\sim\chi^2(n-1)\,$, since they are linear combinations of ...

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