Independence

Question 1

When are random variables X and Y independent

Answer 1

Random variables X and Y are independent when:

FX,Y(x,y) = Fx(x) *FY(y)

Question 2

If RVs X and Y have joint PDF fXY)p(x,y) and have marginal PDFs fX(x) and fY(y), then when are they independent and when does this apply

Answer 2

If RVs X and Y have joint PDF fXY)p(x,y) and have marginal PDFs fX(x) and fY(y), then they are independent when:
fX,Y(x,y) = fX(x) * fY(y)
This applies to more than 2 RVs

Question 3

What is 2D law of unconscious statistician

Answer 3

2D law of unconscious statistician is:
X and Y are continuous RVs and have joint PDFs
E[h(x,y)] = double integral infinity down to - infinity h(x,y) * fX,Y(x,y) dydx
When this is defined

Question 4

If X and Y are jointly continuous 5en what is E[aX +bY]

Answer 4

If X and Y are jointly continuous then :

E(aX + bY) = aE(X) + bE(Y)

Question 5

If X and Y are independent and jointly continuous RVs, what is E(XY)
And E[g(X)h(Y)]

Answer 5

If X and Y are independent and jointly continuous RVs:
E(XY) = E(X)*E(Y)
E[g(X)h(Y)] = E[g(X)] * E[h(Y)]

Question 6

What is definition of covariance of X and Y

Answer 6

Definition of covariance of X and Y is:

E{[X - E(X)][Y - E(Y)]} = E(XY) - E(X)E(Y)

Question 7

What is definition of correlation

Answer 7

Definition of correlation is:

Corr(X,Y) = Cov(X,Y)/root[Var(X)*Var(Y)]

Question 8

What are Cov(X,X) and Cov(X,Y) equal to

Answer 8

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

Question 9

What is Cov(aX +bY,Z) and Var(aX + bY)

Answer 9

Cov(aX +bY,Z) = aCov(X,Z) + bCov(Y,Z)

Var(aX+bY) = a^2Var(X) + b^2Car(Y) + 2abCov(X,Y)

Question 10

When does corr(X,Y) = +-1

Answer 10

Corr(X,Y) = 1 iff Y = aX + B for a > 0
Corr(X,Y) = -1 iff a < 0 and b is real

Question 11

If X and Y are independent, what is Cov(X,Y)

Answer 11

If X and Y are independent:

Cov(X,Y) = 0 BUT COV(X,Y) = 0 DOES NOT IMPLY X AND Y ARE INDEPENDENT