Ch 7

Question 1

What is the joint probability mass function of two discrete variables X and Y?

Answer 1

PX,Y(x,y)=P(X=x,Y=y)

Question 2

What are properties of joint pmfs?

Answer 2

•0<=PX,Y(x,y)<=1 for all x,y
•Sum x,y PX,Y(x,y)=1

Question 3

What is a marginal mass function?

Answer 3

Let X and Y be two discrete RVs with joint pmf. Then the probability mass functions of X and Y are marginal mass functions

Question 4

What are marginal mass functions given by?

Answer 4

pX(x)=sum y PX,Y(x,y) and
pY(y)=sum x PX,Y(x,y)

Question 5

What is joint probability density function of two continuous RVs?

Answer 5

P(X€A,Y€A)=integral A integral B fX,Y(x,y)dydx

Question 6

Properties of joint pdfs?

Answer 6

•greater than 0
•integral over all real =1

Question 7

What are marginal densities given by?

Answer 7

fX(x)=integral R fX,Y(x,y) dy
fY(y)=integral R fX,Y(x,y) dx

Question 8

When are X and Y independent if they have joint density fX,Y?

Answer 8

Iff fX,Y(x,y)=fX(x)fY(y) for all x,y€R

Question 9

Define covariance

Answer 9

Cov(X,Y)=E[XY]-E[X]E[Y]

Question 10

What is cov(X,Y) if X and Y are independent?

Answer 10

0

Question 11

Rewrite Cov(X,Y)

Answer 11

Cov(X,Y)=E[(X-E[X])(Y-E[Y])]

Question 12

When is the covariance positive?

Answer 12

If X and Y tend to increase and decrease together, negative if they do the opposite to each other

Question 13

What are the properties of covariance?

Answer 13

•Cov(X,Y)=Cov(Y,X)
•Cov(X,X)=Var(X)
•Cov(aX*bY,Z)=aCov(X,Y)+bCov(Y,Z) for all a,b€R

Question 14

Define the correlation of two rvs

Answer 14

Cor(X,Y)=(Cov(X,Y))/(sqrt(Var(X)Var(Y))

Question 15

What is Var(X +Y) if C and Y have finite variance?

Answer 15

Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)

Question 16

What is Covariance squared of X and Y if they have finite variance?

Answer 16

(Cov(X,Y))^2<=Var(X)Var(Y)
and -1<=Cor(X,Y)<=1

Question 17

What happens if |Cor(X,Y)|=1?

Answer 17

Y=mX +c with m=Cov(X,Y)/Var(x) and Var(mX-Y)=0