Wednesday, October 29, 2014

correlation coefficient

can 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 $p(x)\mathrm{d}x\propto\exp(-x^4)\mathrm{d}x$?