1 markMCQSection A

Let X be a discrete random variable, then the variance of X is :

Q: Let X be a discrete random variable, then the variance of X is :

The correct option is **(B) $E(X^2) - [E(X)]^2$**. **Reason** - Variance measures spread about the mean: $\text{Var}(X) = E[(X - \mu)^2]$. - Expanding gives the computational form $\text{Var}(X) = E(X^2) - [E(X)]^2$.

Q: How is variance different from standard deviation here?

Variance is $E(X^2) - [E(X)]^2$. Its square root, $\sqrt{E(X^2) - [E(X)]^2}$, is the standard deviation — which is exactly option (D), and therefore not the answer to a question asking for the variance.

Probability Distributions

Variance of a Random Variable

Options

(A)

E(X^2)

(B)

E(X^2) - [E(X)]^2

(C)

E(X^2) + [E(X)]^2

(D)

\sqrt{E(X^2) - [E(X)]^2}

Official Answer

The correct option is (B) $E(X^2) - [E(X)]^2$ .

Reason

Variance measures spread about the mean: $\text{Var}(X) = E[(X - \mu)^2]$ .
Expanding gives the computational form $\text{Var}(X) = E(X^2) - [E(X)]^2$ .

variancediscrete random variableexpectationE(X^2)meansecond momentspreadcomputational formula

Marking Scheme

11 mark: correct option (B) $E(X^2) - [E(X)]^2$ .
2No marks for (A), (C) or (D); the raw second moment, the additive form, or the square-root (standard deviation) form are not accepted.

Hint

Variance = (mean of squares) minus (square of the mean).

Quick Oral Answer

Variance of a discrete random variable is the expected squared deviation from the mean, and its computational form is E(X squared) minus the square of E(X).

Analysis & Explanation

This is the standard short-cut (computational) formula for variance of a discrete random variable.

Concept

By definition $\text{Var}(X) = E[(X - \mu)^2]$ , where $\mu = E(X)$ .
Expanding: $E[X^2 - 2\mu X + \mu^2] = E(X^2) - 2\mu E(X) + \mu^2 = E(X^2) - [E(X)]^2$ .

Why the key is right

Option (B) is exactly this expanded form, so it is correct.

Why the distractors are wrong

(A) $E(X^2)$ alone is the raw second moment, not the variance; it ignores the mean correction term.
(C) uses a plus sign, which over-counts and can never give a spread measure.
(D) $\sqrt{E(X^2) - [E(X)]^2}$ is the standard deviation (the square root of the variance), not the variance itself.

Common Mistakes

1Writing $E(X^2)$ alone and forgetting to subtract $[E(X)]^2$ .
2Adding instead of subtracting the square of the mean.
3Confusing variance with standard deviation by taking the square root, $\sqrt{E(X^2) - [E(X)]^2}$ .

Interesting Facts

The identity $\text{Var}(X) = E(X^2) - [E(X)]^2$ is the discrete analogue of the 'mean of squares minus square of mean' rule used for grouped-data variance in statistics.

Because $\text{Var}(X) \ge 0$ always, this formula proves that $E(X^2) \ge [E(X)]^2$ , a special case of the Cauchy-Schwarz inequality.

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Frequently Asked Questions

Why is variance $E(X^2) - [E(X)]^2$ and not $E(X^2)$ ?

$E(X^2)$ is only the average of the squared values (raw second moment). Variance measures spread about the mean, so the square of the mean, $[E(X)]^2$ , must be subtracted to remove the location and leave only the dispersion.

How is variance different from standard deviation here?

Variance is $E(X^2) - [E(X)]^2$ . Its square root, $\sqrt{E(X^2) - [E(X)]^2}$ , is the standard deviation — which is exactly option (D), and therefore not the answer to a question asking for the variance.