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Part BCSIR NET December 2025the-ratio-needs-zero-mean-and-matched-variance-not-just-independence

The ratio needs zero mean and matched variance not just independence

Let X, Y, and Z be independent Normal random variables with means −1, 0, and 1, respectively, and variances 1, 1, and 3, respectively. Which of the following random variables has a Cauchy distribution with location parameter 0 and scale parameter 1?

  1. A.(X−1)/|Y|
  2. B.(X+2Y+Z)/(X−2Y+Z)✓
  3. C.(Z−1)/Y
  4. D.(X−Y)/(X+Y)

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: If X and Y are each normal and uncorrelated then they are independent

More on this topic

The chapter behind this: Joint distributions, transformations and order statistics — free to read

From Probability › Joint distributions, transformations, order statistics

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