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Part CCSIR NET December 2025the-standard-gumbel-location-is-0-not-ln-ln-2-and-the-density-carries-an-extra-e-to-the-minus-y-factor

The standard gumbel location is 0 not ln ln 2 and the density carries an extra e to the minus y factor

Let {Xn:n≥1X_{n} : n\ge1} be a sequence of independent and identically distributed random variables, where X1X_{1} has an Exponential distribution with mean 1. Define Tn=max⁡T_{n} = \max{X1,X2,…,XnX_{1},X_{2},\dots,X_{n}} − ln n, n≥1. Suppose Tn→T_{n} \toᵈ Y as n→∞n\to\infty. Then which of the following statements are true?

  1. A.Y has a Double Exponential distribution with location parameter ln(ln 2) and scale parameter 1
  2. B.Median of Y = ln(ln 2)
  3. C.P(Y≤0) = e⁻1^{1}✓
  4. D.The derivative of the cumulative distribution function of Y at ln 3 is e^(−1/3)

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: The maximum of independent exponentials is exponential

More on this topic

The chapter behind this: The standard distributions — free to read

From Probability › Standard discrete and continuous distributions

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