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FRM Part I · FRM Exam Part I · Simulation and Bootstrapping

A risk analyst wants to generate draws from a continuous random variable X whose cumulative distribution function F is strictly increasing and invertible. She has a generator producing independent U(0,1) numbers. Which procedure produces draws that follow the distribution of X?

Generate a uniform number U and apply the inverse of the cumulative distribution function, X = F^-1(U). Since F(X) is uniform for a continuous variable, this transformation returns values with exactly the target distribution. Applying F itself or the density to U does not do this.

  1. AGenerate U and compute F(U)
  2. BGenerate U and compute the inverse of F evaluated at UCorrect
  3. CGenerate U and compute 1 minus the density f(U)
  4. DGenerate U and compute the mean of X multiplied by U

Explanation

The inverse transform method sets X = F^-1(U). Because F(X) is uniform on (0,1) for any continuous X, applying the inverse CDF to a uniform draw gives a draw with CDF F. Applying F to U would give a different, generally non-uniform, variable that does not follow X.

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