(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).
(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).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
5
![(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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa54c3a1b-c0c6-4618-b897-4acbc63b0ce8%2Ffc6b5f4e-c183-42c4-a1f0-1c7e391af56d%2F4a4244_processed.png&w=3840&q=75)
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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa54c3a1b-c0c6-4618-b897-4acbc63b0ce8%2Ffc6b5f4e-c183-42c4-a1f0-1c7e391af56d%2F47sjq55_processed.png&w=3840&q=75)
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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