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FRM Part I · FRM Exam Part I · Common Univariate Random Variables

A Bernoulli variable X has success probability p. For which value of p is the variance of X maximized, and what is the maximum variance?

Variance is maximized at p = 0.5, where it equals 0.25. The expression p(1-p) peaks at the midpoint, giving 0.5 x 0.5 = 0.25. At p = 1 the outcome is certain and variance is zero.

  1. Ap = 0.5, variance 0.25Correct
  2. Bp = 0.5, variance 0.50
  3. Cp = 1, variance 1.00
  4. Dp = 0.25, variance 0.1875

Explanation

Variance p(1-p) is a downward parabola with its peak at p = 0.5, where it equals 0.5 x 0.5 = 0.25. At p = 1 the variance is zero because the outcome is certain. At p = 0.25 the variance is 0.1875, which is lower than 0.25.

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