X. Y are random variables. C is a constant. which of the following description about the nature of the mathematical expectation is incorrect?

Question

Options

• AE(C) = CA. E(C) = C
• BE(X+Y) = E(X)+E(Y)
• CE(CX) = CE(X)
• DE(XY) = E(X)E(Y)

• A
  13% (4)
• B
  6% (2)
• C
  3% (1)
• D
  77% (24)

Explanation

Option D is incorrect because E(XY) = E(X)E(Y) is only true when X and Y are statistically independent. Since the problem only states that X and Y are random variables - with no independence assumption - this equality cannot be claimed in general. A simple counterexample: if Y = X, then E(XY) = E(X²), which does not equal E(X)² whenever X has nonzero variance.

Why the other options are correct:

• A (E(C) = C): A constant never varies, so its average is itself - trivially true.
• B (E(X+Y) = E(X)+E(Y)): Linearity of expectation holds for any two random variables, independent or not.
• C (E(CX) = CE(X)): Scaling a variable by a constant scales its expected value - also a direct consequence of linearity.

Memory tip: Think of properties A, B, and C as the "linearity rules" - they hold universally. Option D is the independence trap: the product rule for expectations requires independence, just like the multiplication rule for probabilities (P(A∩B) = P(A)P(B)) does. If you see a product of random variables with no independence stated, be skeptical.

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