Extreme value scaling

• A.WrongBoundary and endpoint

  False, and it is ruled out before any limit is taken.

  The distribution is supported on $[5, \infty)$ — SymPy confirms $F(5) = 1 - e^{0} = 0$ — so every $X_{i}$, and hence every $Y_{n}$, is at least 5.

  The event {$1/2 < Y_{n} < 3/2$} is impossible, with probability 0 for every n.

• B.Correct

  True.

  Each $X_{i}$ is 5 plus an Exp(1) variable

  so $Y_{n} - 5$ is the minimum of n independent Exp(1) variables

  which is Exp(n).

  Then P(|$Y_{n} - 5$| $> \varepsilon) =$ $e^{-n\varepsilon}$ → 0 for every $\varepsilon > 0 ($verified at $\varepsilon = 1/10)$

  which is convergence in probability to 5.

  The trigger: the minimum of n exponentials has rate n

  so it collapses onto the left endpoint at speed 1/n.

• C.WrongExecution slip

  False.

  The scaling is wrong: $Y_{n} - 5$ ~ Exp(n) has size of order 1/n

  so the natural normaliser is n, not $\sqrt{n}$.

  With $\sqrt{n}$ the product has mean $\sqrt{n}/n = 1/\sqrt{n} \to 0 ($verified)

  so $Z_{n} \to 0$ in probability, a degenerate limit.

  And even correctly scaled, $n(Y_{n} - 5)$ ~ Exp(1) — extreme-value limits are exponential or Weibull, never normal.

  $\sqrt{n}$ is the central-limit scaling for sums, not for minima.

• D.WrongExecution slip

  False, for the same reason as (C).

  Since $Z_{n} \to 0$ in probability, $P(1 < Z_{n} < 2) \to 0$, not $\Phi(2) - \Phi(1) \approx 0.136$.

  Any normal-looking answer here signals that a central limit theorem has been applied to a minimum.

3 of these 3 wrong options correspond to a named reasoning error — the same errors recur across subjects.