Wednesday, October 29, 2014

correlation coefficient

can you provide a proof of the Cauchy-Schwarz inequality and show with it that the correlation coefficient
\begin{equation}
r = \frac{\langle xy\rangle}{\sqrt{\langle x^2\rangle\:\langle y^2\rangle}}
\end{equation}
always ranges between $-1\leq r\leq +1$?

bonus question: can you compute the normalisation, variance and kurtosis of $p(x)\mathrm{d}x\propto\exp(-x^4)\mathrm{d}x$?

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