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Part CCSIR NET December 2025cov-of-sum-and-difference-equals-var-x-minus-var-y-it-is-zero-only-when-the-variances-happen-to-match

Cov of sum and difference equals var x minus var y it is zero only when the variances happen to match

Let X and Y be two independent random variables such that the moment generating functions of X and Y are MX(t)=e(3(et−1)),t∈RM_X(t)=e^(3(e^t-1)), t\in\mathbb{R}, and MY(t)=((1/2)e(−3t)+(1/2)e(3t))2,t∈RM_Y(t)=((1/2)e^(-3t)+(1/2)e^(3t))^{2}, t\in\mathbb{R}, respectively. Then which of the following statements are true?

  1. A.P(XY=0) = (1+e⁻3)/2^{3})/2✓
  2. B.E(X+Y) = 3✓
  3. C.Var(X+Y) = 21✓
  4. D.Cov(X+Y,X−Y) = 0

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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Related counterexample: Uncorrelated ⇒ independent

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The chapter behind this: Random variables, moments and generating functions — free to read

From Probability › Random variables, distributions, moments, MGF

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