Suppose we generate two standard uniform samples, 0.15 and 0.5. Which of the following standard normal samples is produced via these two numbers?

Question

BlackTom AI · Accepted Answer

To generate standard normal samples from two independent standard uniform values using the Box-Muller transform, we set U1 = 0.15 and U2 = 0.5.

Compute the common factor: sqrt(-2 * ln(U1)) = sqrt(-2 * ln(0.15)). Since ln(0.15) ≈ -1.897119, we get -2 * ln(0.15) ≈ 3.794238, and its square root is about 1.947.

Then form the two standard normal values:
- Z1 = sqrt(-2 ln(U1)) * cos(2π U2) = 1.947 * cos(2π * 0.5) = 1.947 * cos(π).
  Since cos(π) = -1, Z1 ≈ 1.947 * (-1) ≈ -1.947.
- Z2 = sqrt(-2 ln(U1)) * sin(2π U2) = 1.947 * sin(2π * 0.5) = 1.947 * sin(π).
  Since sin(π) = 0, Z2 ≈ 1.947 * 0 = 0.

Thus, using the two uniform inputs 0.15 and 0.5, the Box-Muller transformation yields two standard normal samples: approximately -1.947 and 0.

If one is asked which standard normal sample is produced via these two numbers, note that both Z1 and Z2 are produced by the pair (0.15, 0.5) in the Box-Muller scheme, with Z1 ≈ -1.947 and Z2 ≈ 0. Although the computed Z1 matches -1.947 precisely to three decimals, Z2 is exactly 0 because sin(π) = 0.

In summary, the pair of standard normal values generated from (0.15, 0.5) under Box-Muller are approximately -1.947 and 0, illustrating that the two uniforms yield two normal samples in one transformation.

25F-STATS-102C-LEC-1 Quiz4- Requires Respondus LockDown Browser

Suppose we generate two standard uniform samples, 0.15 and 0.5. Which of the following standard normal samples is produced via these two numbers?

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