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

An actuary simulates from an Exponential distribution with mean 1 using acceptance-rejection to produce a half-normal variable with density f(x) = sqrt(2/pi) exp(-x^2/2), x > 0, using proposal g(x) = exp(-x). The ratio f/g is maximised at x = 1. What is the bounding constant M, to 3 decimal places?

M = 1.315. The ratio f(x)/g(x) equals sqrt(2/pi) times exp(x - x^2/2), which peaks at x = 1, giving 0.7979 multiplied by e^0.5 = 1.6487, approximately 1.315.

  1. A0.798
  2. B1.253
  3. C1.315Correct
  4. D1.649
  5. 2.000

Explanation

f/g = sqrt(2/pi) exp(x - x^2/2), maximised at x = 1 giving sqrt(2/pi) exp(0.5) = 0.7979 x 1.6487 = 1.315. The value 1.649 omits the sqrt(2/pi) factor; 1.253 is sqrt(pi/2).

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