Skip to content

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

To simulate from a density f(x) = 6x(1 - x) on (0,1) by acceptance-rejection, a Uniform(0,1) proposal is used with the smallest constant M such that f(x) <= M g(x). What is the overall probability that a proposed value is accepted?

The acceptance probability is 2/3. The density peaks at 1.5 at x = 0.5, so the smallest bounding constant against a uniform proposal is M = 1.5, and the chance a proposal is accepted equals 1/M.

  1. A1/3
  2. B2/3Correct
  3. C3/4
  4. D1/2
  5. 1/1.5 squared, i.e. 4/9

Explanation

f is maximised at x = 0.5 where f = 6(0.5)(0.5) = 1.5, so M = 1.5 with g = 1. The acceptance probability is 1/M = 1/1.5 = 2/3. Using M = 3 (a mistaken peak) would give 1/3.

Did you get it right without looking?

One question tells you little. A timed set on Basic univariate distributions and generating samples shows your real accuracy, how long you take and where you lose marks.

Practise Basic univariate distributions and generating samplesFree diagnostic mock

More Basic univariate distributions and generating samples questions