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# P1.T2.713. Uniform, binomial, Poisson distributions (Miller Ch.4)

#### Nicole Seaman

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Learning Objectives: Distinguish the key properties among the following distributions: uniform distribution, Binomial distribution, Poisson distribution.

Questions:

713.1. Let each A_uniform and B_uniform represent independent random uniform variables, where A_uniform falls on the interval between (0, 3) and B_uniform falls on the interval from (4, 10). Which of the following is nearest to the expected standard deviation of X, σ(X), where X = (2*A_uniform + 5*B_uniform)?

a. 0.75
b. 4.90
c. 8.83
d. 12.17

713.2. Peter is analyzing a a granular portfolio that consists of 300 independent and identically distributed (i.i.d.) obligors where each obligor has a default probability of 5.0%. The default pattern of this portfolio is characterized by a binomial distribution. Because he does not have time to compute the binomial, he will use a normal deviate of 2.33 to approximate a 99.0% value at risk (VaR) that employs µ + σ*2.33 to estimate the worst expected number of defaults with 99.0% confidence. This approximation is justified under one rule of thumb that requires n*(1-p) > 10 and n*p >10, which is barely true in this case, as 300*5% = 15.0. Which of the following is Peter's estimate of 99.0% VaR (please note this is technically an absolute VaR as we including the mean, we are not looking for an unexpected loss)?

a. 13 defaults
b. 18 defaults
c. 21 defaults
d. 24 defaults

713.3. A certain low-severity administrative (operational) process tends to produce an average of eight errors per week (where each week is five workdays). If this loss frequency process can be characterized by a Poisson distribution, which is nearest to the probability that more than one error will be produced tomorrow; i.e., Pr(K>1 | λ = 8/5)?

a. 20.19%
b. 32.30%
c. 47.51%
d. 66.49%

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