# correlation coefficient

29 Oct 2014Can you provide a proof of the Cauchy-Schwarz inequality and show with it that the correlation coefficient \(r = \frac{\langle xy\rangle}{\sqrt{\langle x^2\rangle\:\langle y^2\rangle}}\) always ranges between \(-1\leq r\leq +1\)?

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