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Part CCSIR NET December 2023cauchy-no-moments

Cauchy no moments

Suppose U ~ Uniform(0,1) and ½)). Then which of the following statements are true?

  1. A.
  2. B.P(X ∈ {1, 2, 5}) = 1/2
  3. C.E(eˣ) does not exist✓
  4. D.P(X ≤ 0) = 1/2✓

Solution

X is standard Cauchy: symmetric about 0, continuous (so any finite set has probability 0), and with no finite moments — E(eˣ) diverges too.

The trap it tests

Moments and tails

A moment assumed to exist, or tail behaviour assumed to be tame.

Drill statements like this

Related counterexample: Uncorrelated ⇒ independent

More on this topic

From Probability › Random variables, distributions, moments, MGF

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