3.1.2 Joint Cumulative Distribution Functions

Question 1

What is the joint CDF of two random variables X and Y?

Answer 1

F(x, y) = P(X ≤ x, Y ≤ y)

Question 2

How do you compute the joint CDF for discrete random variables?

Answer 2

By double summing the joint PMF:
F(x, y) = Σₐ≤x Σ_b≤y P(X = a, Y = b)

Question 3

What values must a joint CDF fall between?

Answer 3

0 ≤ F(x, y) ≤ 1

Question 4

What does F(x, y) equal when x and y are below their lower bounds?

Answer 4

F(x, y) = 0

Question 5

What does F(x, y) equal when x and y are above their upper bounds?

Answer 5

F(x, y) = 1

Question 6

If x is above the upper bound of X and y is any value, what does F(x, y) simplify to?

Answer 6

It becomes the marginal CDF of Y: F(y)

Question 7

If y is above the upper bound of Y and x is any value, what does F(x, y) simplify to?

Answer 7

It becomes the marginal CDF of X: F_X(x)

Question 8

Can you evaluate F(x, y) at values beyond the defined support of X or Y?

Answer 8

No — it’s not defined or meaningful beyond the support.

Question 9

What is the joint survival function of X and Y?

Answer 9

S(x, y) = P(X > x, Y > y)

Question 10

Is the joint survival function equal to 1 - F(x, y)?

Answer 10

No — unlike the univariate case, joint survival is not equal to 1 - F(x, y)

Question 11

Why can’t you treat joint CDF values like PMF values directly?

Answer 11

Because joint CDF values are cumulative — they sum over regions, not individual outcomes.