What’s the variance of a random variable X where E(X) = 0 ?

Question

BlackTom AI · Accepted Answer

To determine the variance, we start from the standard definition: Var(X) = E[(X - E[X])^2].

Given E[X] = 0, this simplifies to Var(X) = E[(X - 0)^2] = E[X^2].

Therefore, when the mean is zero, the variance equals the second moment E[X^2], provided that E[X^2] exists (i.e., is finite). If E[X^2] is not finite or not defined, the variance is not defined either.

Note on the provided answer: the expression E(X^2) matches the mathematical form Var(X) takes under the condition E[X] = 0, so it correctly represents the variance in this case, assuming E[X^2] is finite. If one were to compute Var(X) without knowing E[X^2], the problem cannot be completed, since the variance depends on that second moment.

In the absence of explicit multiple-choice options in the prompt, this reasoning shows why E[X^2] is the appropriate quantity for Var(X) when the mean is zero.

ERMCPS5350_003_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 6. Quiz for Class 6

What’s the variance of a random variable X where E(X) = 0 ?

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