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Part CCSIR NET December 2024read-the-distribution-off-the-generating-function

Read the distribution off the generating function

Let X and Y be independent random variables with the moment generating functions M_X(t) = (1/9)(2 + eᵗ)2,t∈R)^{2}, t \in \mathbb{R} and M_Y(t) = exp[eᵗ −1],t∈R- 1], t \in \mathbb{R}, respectively. Then, which of the following statements are true?

  1. A.P(XY = 0) = (4 + 5e⁻1)/9^{1})/9✓ Correct
  2. B.E[(3X−Y)2]=6E[(3X - Y)^{2}] = 6✓ Correct
  3. C.Var(X + Y) = 2
  4. D.Cov(2X + Y, X − 2Y) = −10/9✓ Correct

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, and why this one tests standard counterexample.

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The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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Related counterexample: Uncorrelated ⇒ independent

More on this topic

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

From Probability › Random variables, distributions, moments, MGF

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