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IAI Actuarial Core Principles · Actuarial Statistics · Basic univariate distributions and generating samples

A random number U is uniform on (0,1). To generate an exponential claim-interval with mean 5 days by the inverse transform method, which transformation is correct?

The correct transformation is X = −5 ln(U). Inverting the exponential CDF 1 − exp(−x/5) gives x = −5 ln(1 − u), and because 1 − U is also uniform on (0,1), U can replace it, producing positive values with mean 5.

  1. AX = 5U
  2. BX = −5 ln(U)Correct
  3. CX = −ln(U)/5 only if U>0.5
  4. DX = exp(−5U)
  5. X = 5 ln(U)

Explanation

The CDF is F(x)=1−exp(−x/5). Solving u=F(x) gives x=−5 ln(1−u), and since 1−U is also uniform, X=−5 ln U. Option 5 gives negative values. Dividing by 5 treats 5 as the mean's rate.

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