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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Introduction to extreme value theory

Let X1, ..., Xn be independent and identically distributed losses with distribution function F. Which expression gives the distribution function of the maximum Mn = max(X1, ..., Xn)?

The distribution function of the maximum of n independent identically distributed losses is F(x) raised to the power n. The maximum is at most x only when all n losses are at most x, and independence turns this joint event into a product.

  1. An F(x)
  2. BF(x)^nCorrect
  3. C1 - (1 - F(x))^n
  4. D1 - F(x)^n
  5. F(x)/n

Explanation

The maximum is at most x only if every Xi is at most x. By independence, P(Mn ≤ x) = F(x)^n. Option 1 - (1 - F(x))^n is the distribution function of the minimum, which is the common confusion.

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