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

A risk manager considers 10 independent loans, each defaulting with probability 0.10 modeled as identical Bernoulli trials. Let Y be the number of defaults. Which statement about Y and the Bernoulli building block is correct, and what is P(Y = 0)?

Y is binomial with n = 10 and p = 0.10, being a sum of independent Bernoulli trials. The probability of no defaults is every loan surviving, 0.9^10, about 0.349. Other options mislabel the distribution or compute the chance that not all loans default.

  1. AY is binomial; P(Y = 0) = 0.9^10 = 0.349Correct
  2. BY is Bernoulli; P(Y = 0) = 0.90
  3. CY is binomial; P(Y = 0) = 1 - 0.1^10 = 0.9999999999
  4. DY is binomial; P(Y = 0) = 10 x 0.9 x 0.1^9

Explanation

The sum of n independent identical Bernoulli variables is binomial(10, 0.10). P(Y=0) requires all 10 to survive: 0.9^10 = 0.3487. Option 3 gives the probability that not all default. Option 2 mislabels Y as Bernoulli.

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