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

The random variable X has cdf F(x) = 1 - (2/x)^3 for x >= 2. Using the inverse transform method, what value of X results from a Uniform(0,1) draw u = 0.875?

X = 4. Solving 1 - (2/x)^3 = 0.875 gives (2/x)^3 = 0.125, so 2/x = 0.5 and x = 4. The inverse transform simply sets the cdf equal to the uniform draw and inverts it.

  1. A3.5
  2. B4Correct
  3. C8
  4. D2.5
  5. 2.857

Explanation

Set 1 - (2/x)^3 = u, so (2/x)^3 = 1 - u = 0.125. Then 2/x = 0.5, giving x = 4. Using u instead of 1-u would give (2/x)^3 = 0.875, x about 2.09, not offered; 8 results from forgetting the cube root.

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