Joint Distributions

Question 1

When are X and Y jointly continuous RVs

Answer 1

X and Y are jointly continuous RVs when:
Exists a function R^2 to [,infinity) s,t for any region A
P((X,Y) in A) = double integral over A fX,Y(x,y) dydx
Function is joint PDF of X and Y

Question 2

What is joint CDF defined by

Answer 2

Joint CDF is defined by:
Function R^2 to [0,1] FX,Y(x,y) = P(X <= x and Y <= y)
= double integral x down to -infinity y down to -infinity fX,Y(u,v) dv du

Question 3

What are marginal PDFs

Answer 3

Marginal PDFs are:
If X and Y have joint PDF fX,Y(x,y) then X and Y have PDFs
fx = integral infinity down to -infinity fX,Y(x,y) dy
fy = integral infinity down to -infinity fX,Y(x,y) dx
These are marginal PDFs of X and Y

Question 4

What is P(Y=y|X=x) for discrete RVs

Answer 4

For discrete RVs,

P(Y=y|X=x) = P(Y=y, X =x)/P(X=x) where P(X=x) > 0

Question 5

What is conditional PDF of Y given X=x

Answer 5

Conditional PDF of Y given X=x is:

fY|X (y|x) = fY|X (y|X=x)= fX,Y(x,y)/fX(x) where fX(x) > 0 and y is real

Question 6

What is P(a <= Y <= b|X=x)

Answer 6

P(a <= Y <= b|X=x) = integral b down to a fY|X (y|x) dy

Question 7

What is conditional CDF of Y given X=x

Answer 7

Conditional CDF of Y given X=x is:

FY|X (y|x) = P(Y <= y|X=x) = integral y down to - infinity fY|X(u|x) du

Question 8

What is conditional expectation of Y|X=x

Answer 8

Conditional expectation Y|x is:

E(Y|X=x) = integral infinity down to -infinity y* fY|X(y|x) dy when this exists

Question 9

What does the multiplication rule state

Answer 9

Multiplication rule states:

FX,Y(x,y) = fX(x) * fY|X (y|x) = fY(y) * fX|Y(x|y)

Question 10

What is independence property for conditional PDFs

Answer 10

Independence property for conditional PDFs is:

fY,X (y|x) = fY(y) for all y in R