Part CCSIR NET December 2024the-transform-is-uniform-not-cauchy

The transform is uniform not cauchy

Let X be a random variable with probability density function $f (x) = 1/ (π (1 + x^{2})), x \in R$ . If $Z = 1/2 + (1/ π) tan$ ⁻ $^{1} (X)$ , then which of the following statements are true?

A.E(Zᵐ) = 1/(m + 1), for all $m \in N$ ✓
B. $E (Φ$ ⁻ $^{1} (Z)) = 0$ , where $Φ (\cdot)$ is the cumulative distribution function of standard normal random variable.✓
C.Z is degenerate at 0.
D.If $Z_{1}$ and $Z_{2}$ are independent and identically distributed (i.i.d.) random variables having distribution same as the distribution of Z, then Z has the same distribution as $(Z_{1} + Z_{2}) /2$ .

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.

See pricing

50 are analysed free — try those first.

Related counterexample: Uncorrelated ⇒ independent

The transform is uniform not cauchy

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