Questions: Joint Distributions and Marginals (Rigorous)

Question 1

Random variables X and Y each have marginal distributions that are Uniform[0,1]. Which of the following conclusions necessarily follows?

• A. The joint pdf must be f(x,y) = 1 on the unit square — they must be independent
• B. The joint CDF is fully determined by the two marginals
• C. The joint distribution could be any of infinitely many possibilities — independent, positively correlated, negatively correlated, and more — all consistent with these marginals
• D. X and Y must have zero covariance because they share the same marginal

Answer: (C)

Marginal distributions describe each variable in isolation; they do not determine how the variables relate to each other. Two random variables can share identical marginals while being independent, perfectly positively correlated, perfectly negatively correlated, or anything in between. The joint distribution encodes the dependence structure, which the marginals alone cannot recover. This is the central insight of the topic: the joint contains strictly more information than the marginals together.

Question 2

To rigorously verify that two continuous random variables X and Y are independent, you must show:

• A. Their means and variances are equal
• B. The joint pdf factors as f(x,y) = f_X(x) · f_Y(y) for (almost) all (x,y)
• C. Their marginal pdfs each integrate to 1
• D. Their covariance equals zero

Answer: (B)

Independence is precisely the condition that the joint pdf factors into the product of the marginals. This factorization must hold across all (x,y), not just at selected points. Note that zero covariance (option D) is *necessary* for independence but not sufficient — two dependent variables can have zero covariance (e.g., if Y = X²). The factorization of the joint pdf is the rigorous definition and the primary verification tool.

Question 3

If two random variables have identical marginal distributions, they should have the same joint distribution.

• T. True
• F. False

Answer: False

This is the most important misconception this topic addresses. Shared marginals do not determine the joint distribution. A standard counterexample: X and Y both Uniform[0,1] as marginals, but (X,Y) could be jointly uniform on the unit square (independent) or supported only on the diagonal y = x (perfectly correlated) — the marginals are the same in both cases, but the joint distributions are completely different.

Question 4

The marginal pdf of X₁ from a continuous joint distribution can be obtained by integrating the joint pdf f(x₁, x₂) over all values of x₂.

• T. True
• F. False

Answer: True

This is the defining operation for marginals: f₁(x₁) = ∫₋∞^∞ f(x₁, x₂) dx₂. The integration 'sums over' all possible values of the second variable, collapsing the two-dimensional distribution onto one axis. Fubini's theorem, applicable under the absolute continuity assumptions that justify the existence of the joint density via Radon-Nikodym, guarantees this integration is well-defined and can be performed in either order.

Question 5

Why does knowing both marginal distributions of a random vector (X, Y) not give you the complete probabilistic story of (X, Y)?

Model answer: The marginals describe each variable in isolation — X's distribution ignoring Y, and Y's distribution ignoring X. They contain no information about how the two variables relate to each other. The joint distribution encodes whether large X values tend to coincide with large Y values, whether the variables are independent, and so on. Two distributions can have identical marginals while being completely independent or tightly correlated; only the joint distribution distinguishes them.

The point generalizes: in statistics, knowing the marginal distributions of two (or more) variables is often insufficient for prediction, simulation, or inference about joint events. The joint distribution — and specifically the dependence structure it encodes — is what matters for questions like 'what is the probability that both X > a and Y > b?' Marginals only answer questions about each variable separately.