Question

The (population) correlation coefficient, called p, is discussed in Section 4.5.2 of your text. Given two random variables X and Y with some joint distribution and means ux and uy, p= Corr(X,Y) = Cov(X, Y), where σχσY oſ = Var(x), of = Var(y) and Cov(X,Y) = E[(X - MX)(Y – My)] Given data, we can estimate p. Suppose that (X1,Y1), ..., (Xn, Yn) are independent and iden- tically distributed (i.i.d.) pairs of realizations of the random variables (X, Y).

Answer 1

The simple, tinear suegression Slope is defined :- as, and the correlation coefficient is given by to* fta, y) 61) Trening - the above formula gives the exact relationship. between the two meascere of Statisticoul relationship two Simple univariate data The regression slope gives the "best" linear approximation of the relationship between y; and a; whereas the correlation coefficient tells us how linearly predictable y; is given a value of ni. But regression slope is unbounded, and it be real number, but the correlation coefficient lies between [1,1] suppose X and Y are random variables suppose also that we would like to fit limar regression line y using x. : Say = covex,y) Sex = cov (x,x) - Varcx) Y = Bot Bit is the model we will use then i cor(x, y) Var (x)

The larger il is, the better least square fit pe o R? gives the percentage variation in y explained by X-, variables. The range is o to la we multiply this percentage of variation to the ratio of two variances (ix and y) gives