Cauchy no moments

• A.WrongMoments and tails

  False.

  X is standard Cauchy, and the Cauchy distribution has no finite moments of any positive order.

  SymPy samples the fourth-moment integrand $x^{4}/(\pi(1 + x^{2}))$ and finds it growing without bound — it behaves like $x^{2}/\pi$ for large x

  so the integral diverges.

  Not even E(X) exists, let alone $E(X^{4})$.

• B.Wrong

  False.

  X has a continuous distribution — SymPy confirms the density integrates to 1 over $\mathbb{R}$ — so every finite set has probability exactly 0.

  A probability of 1/2 on three isolated points would require atoms

  which a density rules out.

• C.Correct

  True.

  E(eˣ$) = \int e$ˣ$/(\pi(1 + x^{2}))$ dx over $\mathbb{R}$, and SymPy samples that integrand from x = 1 to 25, where it explodes: the exponential in the numerator overwhelms the quadratic denominator.

  The integral diverges at $+\infty$

  so the expectation does not exist.

  The trigger: Cauchy tails decay like $1/x^{2}$

  which is too slow to tame any growing function.

• D.Correct

  True.

  U − ½ is symmetric about 0 on (−½, ½), and tan is an odd function

  so X is symmetric about 0.

  Hence P(X ≤ 0) = 1/2 — SymPy confirms $\int_{-\infty}^{0}$ of the standard Cauchy density is exactly 1/2.

  Symmetry survives even though no moments do; the median exists where the mean does not.

1 of these 2 wrong options correspond to a named reasoning error — the same errors recur across subjects.