Cov(2X, X) is equal to

Question

BlackTom AI · Accepted Answer

Consider the question: Cov(2X, X) is equal to what?

First, recall a basic property of covariance: Cov(aU, V) = a Cov(U, V) for any constant a. Applying this with a = 2, U = X, and V = X gives Cov(2X, X) = 2 Cov(X, X).

Next, Cov(X, X) is by definition the variance of X, Var(X).

Putting these together yields Cov(2X, X) = 2 Var(X).

Now evaluate the provided options:
- Option 1: 2*Var(X) — This matches the derived result, so it is correct.
- Option 2: 4*Var(X) — This would imply Cov(2X, X) = 4 Var(X), which double-counts the scaling effect and is incorrect.
- Option 3: 2*Corr(X) — Covariance is not expressed directly in terms of correlation without involving standard deviations; this option confuses covariance with correlation and is incorrect.
- Option 4: 4*Corr(X) — Similar to Option 3, this mixes a covariance scaling with correlation, which is not valid; this is incorrect.

In summary, the correct relationship is Cov(2X, X) = 2 Var(X), and the other choices misapply scaling or confuse covariance with correlation.

ERMCPS5350_003_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 8. Quiz for Class 8

Cov(2X, X) is equal to

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