 Hello and welcome to the session. In this session, first of all, let us discuss properties of regression coefficients. And the first property is correlation coefficient is the genetic mean between the regression coefficients. Now the regression coefficients b y x which is equal to r into sigma y over sigma x and b x y which is equal to r into sigma x over sigma y where b y x is the regression coefficient of y on x and b x y is the regression coefficient of x on y is the correlation coefficient is the standard deviation of y and sigma x is the standard deviation of x. Now b y x into b x y is equal to r into sigma y over sigma x into r into sigma x over sigma y which further into b x y is equal to this further is r square root of b y x into b x y. Thus we can say that the correlation coefficient r is the regression mean between the two regression coefficients b y x and b x y. Now from this equation v r is equal to b y x into d x y which is further equal to of d x into d y over into into d x into d y over is equal to summation of d x into d y over sigma x into sigma y which is called correct moment formula. Now let us start with the second property and which is if one of the regression coefficients is greater than unity numerically the other must be less than unity numerically. Now the regression coefficients are b y x and b x y. Now let the regression coefficient of x y is greater than or equal to 1 then this implies 1 over b x y is less than equal to 1. Now let this be equation number 1 b y x into b x y is equal to r square this we have proved earlier so b y x into b x y is equal to r square is less than equal to 1. Therefore this is less than equal to 1 over b x y is less than equal to 1 and this is from this equation that is from the equation number 1 1 over b x y is less than equal to 1. So we have taken one of the regression coefficients that is b x y is greater than equal to 1 and we are letting the other regression coefficient that is b y x is less than equal to mean of the regression coefficients. Now we know what does it mean the geometric mean of the given series. Now for the regression coefficients b y x and b x y the alpha is greater than the geometric mean and that is b x y regression coefficient that is because r is equal to b x y. Therefore we can say that the arithmetic mean of the regression coefficient is greater than which is r. Now let us discuss the next property and that is the correlation coefficient in coefficients that is b y x where r is the correlation coefficient. Sigma y is the standard deviation of y and sigma x is the standard deviation of n that is b x y is equal to r into r both positive have the same sum b y x and b x y both are positive and if r is negative then b x y both are negative. Now let us discuss the next property and which is the regression coefficient. Now here let u is equal to this b over k that is u is equal to x minus a over h and b is equal to y y x that is b y x is equal to sigma y over sigma x will be equal to r. Now sigma y will be equal to k into sigma v whole upon sigma x will be equal to sigma u which is further written as k over h into sigma v over sigma u which is equal to k over h into v u. That is b v u is the regression coefficient of v over u over k into now here b x y is equal to h by k into b u v that is the regression coefficient of h of u over v. Now the regression equations y bar is equal to r into sigma y over sigma x into x minus x by the whole r is equal to r into sigma x over sigma y into y minus y by the whole. As the equation of regression the slopes that is the slope m1 is equal to r into sigma y over sigma x that is equation of y of x which is given by equation number 1 to sigma x. Question number 2 between the line that is the regression line 1 then is equal to 1 plus m1 m2 plus minus sigma y over sigma x minus sigma y over sigma x whole upon sigma x. On solving this this will be equal to plus minus into sigma x into sigma y whole upon therefore sin gives the equation between the lines. We have got this as the value of that sigma x into sigma y whole upon sigma x square is we are considering only the positive sign every one that is when r is equal to 0 then equal to infinity and this is like infinity which is equal to equation of regression are perpendicular. The estimated value now let us discuss the second corollary and that is when r is equal to plus minus 1 equal to 0 implies theta is equal to 0 think theta is 0 regression lines is 0 or pi then coincide and there is a perfect correlation between the two lines of regression. So this completes our session hope you all enjoyed this video. Thank you.