(a) Show that Cov(X,Y)=0 if X,Y are independent. Hint: Find a computational formula for covariance, similar to the com- putational formula for variance, Var(X)= E(x²)– [E(X)]². (b) Show, by means of an example, that Cov(X,Y)=0 does not imply that X,Y are independent. (c) Show that Var(X +Y) = Var(X)+Var(Y)+2· Cov(X,Y).

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(a) Show that Cov(X,Y)=0 if X,Y are independent.
Hint: Find a computational formula for covariance, similar to the com-
putational formula for variance, Var(X)=E(x²)– [E(X)]².
(b) Show, by means of an example, that Cov(X,Y)=0 does not imply that
X,Y are independent.
(c) Show that
Var(X +Y) = Var(X)+Var(Y)+2·Cov(X,Y).
(d) Show that [E(XY)]² < E(X)E(Y).
Hint: Let Z = X +aY, where a E R. Note that E(Z?) > 0.
(e) The Pearson correlation coefficient
Cov(X,X)
Var(X)Var(Y)
P(X,Y)=
measures the linear relationship between X and Y. Using part (d),
deduce that –1 <p(X,Y)<1.
Note: If p > 0 we say that X, Y are positively correlated. Likewise, if
p < 0 we say that X,Y are negatively correlated.
Note: It can be shown (you do not need to) that if p =±1 then the
relationship is perfectly linear, Y = aX +b, for some a,b e R, with
a < 0 if p = -1 and a > 0 if p =+1.
%3D
Transcribed Image Text:(a) Show that Cov(X,Y)=0 if X,Y are independent. Hint: Find a computational formula for covariance, similar to the com- putational formula for variance, Var(X)=E(x²)– [E(X)]². (b) Show, by means of an example, that Cov(X,Y)=0 does not imply that X,Y are independent. (c) Show that Var(X +Y) = Var(X)+Var(Y)+2·Cov(X,Y). (d) Show that [E(XY)]² < E(X)E(Y). Hint: Let Z = X +aY, where a E R. Note that E(Z?) > 0. (e) The Pearson correlation coefficient Cov(X,X) Var(X)Var(Y) P(X,Y)= measures the linear relationship between X and Y. Using part (d), deduce that –1 <p(X,Y)<1. Note: If p > 0 we say that X, Y are positively correlated. Likewise, if p < 0 we say that X,Y are negatively correlated. Note: It can be shown (you do not need to) that if p =±1 then the relationship is perfectly linear, Y = aX +b, for some a,b e R, with a < 0 if p = -1 and a > 0 if p =+1. %3D
5. The covariance of random variables X,Y is defined as
Cov(X,Y)=E[(X – µx)(Y – µy)]
where ux = E(X) and µy = E(Y).
Note: Var(X) = Cov(X,X).
Transcribed Image Text:5. The covariance of random variables X,Y is defined as Cov(X,Y)=E[(X – µx)(Y – µy)] where ux = E(X) and µy = E(Y). Note: Var(X) = Cov(X,X).
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