Questions: Bivariate Normal Distribution

Question 1

For a bivariate normal distribution with ρ = 0, X and Y are uncorrelated. What additional conclusion can you draw that would NOT hold for a general joint distribution?

• A. X and Y have identical marginal distributions
• B. X and Y are statistically independent
• C. The joint density is constant along circles centered at (μ₁, μ₂)
• D. The conditional variance of Y given X is zero

Answer: (B)

For the bivariate normal specifically, ρ = 0 implies full independence — a much stronger statement than for distributions in general. In general, uncorrelated random variables can still be dependent. A classic example: X ~ Uniform(−1, 1) and Y = X² have zero correlation but Y is completely determined by X. Zero correlation measures only linear association; for the bivariate normal, all dependence is linear, so zero linear association means no association at all.

Question 2

In a bivariate normal distribution with ρ = 0.9, how do the contour ellipses of the joint density compare to those when ρ = 0?

• A. They are rounder — high correlation compresses the distribution symmetrically
• B. They are larger but have the same axis alignment
• C. They are tilted and elongated, approaching a line as ρ → 1
• D. They are identical — ρ affects only the conditional mean, not the geometry

Answer: (C)

When ρ = 0 the contour ellipses are axis-aligned. As |ρ| increases, the ellipses tilt (ρ > 0 tilts northeast, ρ < 0 tilts southeast) and elongate. At ρ = ±1 the ellipses collapse to a line, corresponding to a perfect linear relationship. The correlation parameter ρ directly controls the orientation and elongation of these ellipses — it is geometrically encoded in the angle and eccentricity of the contours.

Question 3

For any joint probability distribution, if two random variables have zero correlation, they are independent.

• T. True
• F. False

Answer: False

Zero correlation implies independence only for the bivariate normal, not in general. Counterexample: let X ~ Uniform(−1, 1) and Y = X². Then Cov(X, Y) = 0 by symmetry, so ρ = 0, yet Y is completely determined by X. Zero correlation measures linear association only. The bivariate normal's special property is that all dependence in the distribution is captured by ρ, so when the linear association is zero, all association is zero.

Question 4

In a bivariate normal distribution, the conditional distribution Y|X = x is itself a normal distribution.

• T. True
• F. False

Answer: True

This is one of the defining properties of the bivariate normal: all conditional distributions are normal. Specifically, Y|X = x ~ N(μ₂ + ρ(σ₂/σ₁)(x − μ₁), σ₂²(1 − ρ²)). The conditional mean is a linear function of x — the population regression line — and the conditional variance is constant across all values of x. Normal marginals, normal conditionals, and linear conditional means are the signature of the bivariate normal family.

Question 5

How does the correlation parameter ρ affect the conditional distribution of Y given X = x in a bivariate normal? What happens to the conditional variance as |ρ| → 1, and what does this mean geometrically?

Model answer: The conditional distribution Y|X = x is normal with mean μ₂ + ρ(σ₂/σ₁)(x − μ₁) and variance σ₂²(1 − ρ²). As |ρ| → 1, the conditional variance approaches 0 — given X, Y is almost perfectly predictable. Geometrically this corresponds to the contour ellipses collapsing toward a line: when ρ = ±1, Y is an exact linear function of X.

The conditional mean shows ρ's effect on the regression relationship — stronger correlation means knowing X shifts the center of Y's distribution substantially. The conditional variance σ₂²(1 − ρ²) is the residual uncertainty after observing X; it shrinks toward 0 as |ρ| → 1. At ρ = 0, the conditional variance equals the marginal variance σ₂² (X tells you nothing about Y). This is the population version of R² = ρ² in simple linear regression: ρ² of the variance in Y is explained by X.