X, Y are random variables, C is a constant, the nature of the difference in the following options which is wrong? (Multiple choice)

Question

Options

• AD(C)=O
• BD(X+ Y)=D(X)+D(Y)
• CD(CX)=C'C'D()()
• DD(XY)=D(X)D{Y)

• A
  10% (5)
• B
  73% (37)
• C
  18% (9)

Explanation

Options B and D are wrong because they state equalities that only hold under special conditions, not in general. D(X+Y) = D(X) + D(Y) is only valid when X and Y are independent (or at least uncorrelated); the correct general formula is D(X+Y) = D(X) + D(Y) + 2Cov(X,Y), where the covariance term is omitted in B. D(XY) = D(X)D(Y) has no general validity at all - the variance of a product involves higher-order moments and does not factor this cleanly under any standard condition. Options A and C are correct: D(C) = 0 because a constant has no variability, and D(CX) = C²D(X) because the constant scales deviations and variance squares that scaling factor.

Memory tip: For variance, constants kill spread (A), scalar multiplication squares (C), and anything involving sums or products of two random variables requires checking independence - if independence is not given, formulas B and D both fail.

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