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Part CCSIR NET December 2023shifted-cauchy-sign

Shifted cauchy sign

Suppose X is a continuous random variable with density f(x)=(1/π)⋅1/(1+(x+1)2)f(x) = (1/\pi)\cdot1/(1 + (x + 1)^{2}), and Y = X/|X| for X ≠ 0, Y = 0 for X = 0. Then which of the following statements are true?

  1. A.E(Y) = 0
  2. B.P(Y > 0) < P(Y < 0)✓ Correct
  3. C.P(Y < −1) < P(Y > 1)
  4. D.E(Y2)=1E(Y^{2}) = 1✓ 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 boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

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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