Prove that coefficient of correlation always lies between 1 and -1. Please do not give links. Share with your friends Share 1 Vaishali Nebhwani answered this Dear student, coefficient of correlation = r = cov (x,y)σxσyσx2=Σ(y-x¯)2ncov (x,y)=Σ(x-x¯)(y-y¯)nwe have to prove -1≤r≤1Let's consider two terms x-x¯σx and y-y¯σytaking sum of squares of both termsΣx-x¯σx ± y-y¯σy2(given no is positive since its square)⇒Σx-x¯σx ± y-y¯σy2 ≥0opening square we getΣx-x¯σx2 + y-y¯σy2+2x-x¯σx y-y¯σy ≥0on simplifying,Σx-x¯2σ2x+Σy-y¯2σ2y±2Σx-x¯y-y¯σxσy≥0diving by n Σx-x¯2nσ2x+Σy-y¯2nσ2y±2Σx-x¯y-y¯nσxnσy≥0using formula of variance,σ2xσ2x+σ2yσ2y±2covx,yσxσy≥01+1±2r≥02+2r≥01≥-rr≥-1or 2-2r≥02≥2r1≥r⇒-1≤r≤1hence proved Regards -1 View Full Answer
