Questions: Covariance and Correlation of Random Variables

Question 1

Let X be uniformly distributed on [−1, 1] and let Y = X². What is Cov(X, Y)?

• A. Positive — because when X is large, Y = X² is also large
• B. Negative — because Y = X² is a decreasing function of X when X < 0
• C. Zero — despite Y being completely determined by X
• D. Undefined — covariance requires both variables to have variance greater than zero

Answer: (C)

Cov(X, Y) = E[XY] − E[X]E[X²] = E[X³] − 0 · E[X²] = 0, because X is symmetric around 0, so E[X³] = 0. Yet Y = X² is completely determined by X — perfect nonlinear dependence. This is the key illustration that zero covariance (and zero correlation) does NOT imply independence. Covariance only measures linear association; the relationship between X and X² is purely nonlinear (a parabola), which is invisible to covariance. Option A and B are both tempting because the relationship between X and X² is real — covariance just can't see it.

Question 2

A dataset shows Cov(height_cm, weight_kg) = 450 and Cov(height_m, weight_kg) = 4.5. What does this comparison illustrate?

• A. The relationship between height and weight is 100 times stronger when measured in centimeters
• B. Covariance depends on the units of measurement, so its magnitude alone cannot indicate the strength of a relationship
• C. The correlation between height and weight is also 100 times larger when using centimeters
• D. The dataset with higher covariance has measurement error — both should give the same covariance

Answer: (B)

Covariance is not scale-free: Cov(aX, Y) = a · Cov(X, Y). Converting height from meters to centimeters multiplies by 100, which multiplies the covariance by 100 — but the underlying relationship between height and weight hasn't changed. This is exactly why correlation was invented: ρ = Cov(X,Y)/(σ_X σ_Y) normalizes by the standard deviations, producing a dimensionless measure in [−1, 1] that doesn't change when you rescale either variable. Option C is wrong: correlation is unit-free, so it remains the same regardless of whether height is in meters or centimeters.

Question 3

If X and Y are independent random variables, then Cov(X, Y) = 0.

• T. True
• F. False

Answer: True

Independence means E[XY] = E[X]E[Y], which directly gives Cov(X, Y) = E[XY] − E[X]E[Y] = 0. This direction is always true. Independence implies zero covariance. The practical consequence is that Var(X + Y) = Var(X) + Var(Y) when X and Y are independent — the covariance cross-term vanishes. This is the result used constantly in probability theory, statistics, and portfolio theory.

Question 4

If Cov(X, Y) = 0, then X and Y are independent.

• T. True
• F. False

Answer: False

Zero covariance does not imply independence — this is the most important misconception about covariance. The converse of 'independence ⟹ zero covariance' does not hold. The canonical counterexample: X uniform on [−1, 1], Y = X². Here Cov(X, Y) = 0, but Y is a deterministic function of X — perfect dependence. Covariance only measures linear association, so any nonlinear relationship (quadratic, circular, etc.) can produce zero covariance while X and Y remain completely dependent.

Question 5

Explain why zero correlation between two random variables does not imply that they are independent, and give an example.

Model answer: Correlation measures only linear association — how much X and Y vary together in a straight-line pattern. A relationship that is nonlinear can produce zero correlation even when one variable completely determines the other. For example, let X ~ Uniform(−1, 1) and Y = X². Then Cor(X, Y) = 0 because the relationship is symmetric (X = 1 and X = −1 both give Y = 1, so positive and negative deviations of X cancel in the covariance), yet Y is a deterministic function of X — knowing X tells you Y exactly.

The independence condition requires E[f(X)g(Y)] = E[f(X)]E[g(Y)] for all functions f and g. Correlation only checks this for the specific functions f(x) = x and g(y) = y. All the other function pairs are ignored. If the dependence between X and Y is captured by a nonlinear relationship, it won't appear in the covariance or correlation at all.