The coefficient of correlation between two variable x and y is given by
r=[σx2+σy2+σx-y2]/[2σxσY]
r=[σx2+σy2-σx-y2]/[2σxσY]
r=[σx2+σy2+σx-y2]/[σxσY]
r=[σx2+σy2-σx-y2]/[σxσY]
Explanation for the correct option :
Finding relation between r and σx-y2:
σx-y2=[(x-y)-(x¯-y¯)]2n=[(x-x¯)-(y-y¯)]2n=(x-x¯)2+(y-y¯)2-2(x-x¯)(y-y¯)n=(x-x¯)2n+(y-y¯)2n-2(x-x¯)(y-y¯)n=σx2+σy2-2Cov(x,y)=σx2+σy2-2rσxσy
Rearrange above equation.
∴2rσxσy=σx2+σy2-σx-y2⇒r=σx2+σy2-σx-y22σxσy
Therefore, option (B) is correct.