Moments and tails

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

Would you have fallen for these?

Each is a real option from a previous-year paper. Read it cold and decide before opening it.

• Option A

  $E(X^4)=3$

  Why this is wrong, and why it looks right

  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 $E(X^4)$.

  From this question

  Suppose U ~ Uniform(0,1) and $X=\tan (\pi (U-$ ½)). Then which of the following statements are true?

  See the full question

  Part C · december 2023 · free

• Option A

  There exists an $\alpha \in (0,\infty )$ such that $li{m}_{x\to {\alpha }^{-}}$ |y(x)| $=\infty$.

  The analysis of this one comes with the PYQ pack.

  Part C · december 2023

Other mistakes that cost marks