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Official PYQ · 2011Q18Probability: Random Variables and DistributionsMCQ1 mark

GATE CS 2011 Q18 — Probability: Random Variables and Distributions

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If the difference between the expectation of the square of a random variable (E[X2])\left(E\left[X^2\right]\right) and the square of the expectation of the random variable (E[X])2\left(E\left[X\right]\right)^2 is denoted by RR, then

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Answer: C

By definition, the variance of a random variable XX, denoted by Var(X)\text{Var}(X), is given by:

Var(X)=E[(X−E[X])2]\text{Var}(X) = E\left[(X - E[X])^2\right]

Expanding this expression using the linearity of expectation, we get:

Var(X)=E[X2−2XE[X]+(E[X])2]\text{Var}(X) = E\left[X^2 - 2X E[X] + (E[X])^2\right] Var(X)=E[X2]−2E[X]E[X]+(E[X])2\text{Var}(X) = E[X^2] - 2E[X]E[X] + (E[X])^2 Var(X)=E[X2]−(E[X])2\text{Var}(X) = E[X^2] - (E[X])^2

The problem defines this quantity as RR:

R=E[X2]−(E[X])2=Var(X)R = E[X^2] - (E[X])^2 = \text{Var}(X)

Since (X−E[X])2≥0(X - E[X])^2 \ge 0 for all real values of XX, its expected value must also be non-negative:

E[(X−E[X])2]≥0  ⟹  R≥0E\left[(X - E[X])^2\right] \ge 0 \implies R \ge 0

Specifically:

  • R=0R = 0 if and only if XX is a constant (a degenerate random variable with zero spread).
  • R>0R > 0 for any non-constant random variable.

Thus, in the general case, R≥0R \ge 0.

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