FRM Part I · FRM Exam Part I · Common Univariate Random Variables
A portfolio holds 3 independent loans, each modeled by a Bernoulli variable with default probability 0.10. Using the sum of the three Bernoulli variables, what is the expected number of defaults and its variance?
The expected number of defaults is 0.30 and the variance is 0.27. Each loan has mean 0.10 and variance 0.09, and with independence both add across three loans. Using a single loan's variance of 0.09 would be wrong.
- AMean 0.30, variance 0.09
- BMean 0.30, variance 0.27Correct
- CMean 0.10, variance 0.09
- DMean 0.27, variance 0.30
Explanation
The mean of each is 0.10, so the sum has mean 3 x 0.10 = 0.30. Each variance is 0.10 x 0.90 = 0.09, and independence lets variances add: 3 x 0.09 = 0.27. The 0.09 variance ignores summing across loans.
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