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P1.T2.20.5. Common univariate random variables (uniform, Bernoulli, binomial, and Poisson)

Nicole Seaman

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Learning objectives: Distinguish the key properties and identify the common occurrences of the following distributions: uniform distribution, Bernoulli distribution, binomial distribution, Poisson distribution ...


20.5.1. Assume two continuous uniform random variables, X ~ U(a1, b1) and Y ~ U(a2, b2), except we are told their support starts with zero such that we know X ~ U(0, b1) and Y ~ U(0, b2) so we know they share a lower bound of zero but we do not know their respective upper bounds. We are told the variance of the first variable (X) is 12.0 and the variance of the second variable (Y) is 27.0. If the variables are independent, what is the joint probability that both will be greater than nine; i.e., Pr(X > 9 ∩ Y >9)?

a. Zero
b. 12.50%
c. 25.00%
d. 50.00%

20.5.2. A credit portfolio contains some number of independent credit-sensitive assets with identical default probabilities; as the defaults are i.i.d., we can use the binomial distribution to characterize the number of defaults. We are told the expected number of defaults is 4.0 with a variance of 3.80. Which is nearest to the probability of exactly four defaults; Pr(X = 4 | binomial with mean of 4.0 and variance of 3.80)?

a. Less than 0.01%
b. 10.0%
c. 20.0%
d. 33.3%

20.5.3. On average, one bot hits a website every 12.0 seconds. If we can characterize this frequency during a time interval with the Poisson distribution (i.e., the key assumption is that the events--the bot hits--are independent), then which of the following is nearest to the probability of not more than one bot hit (≤ 1.0; i.e, less than or equal to one bot hit) over a one minute (60-second) interval.

Put another way, what is the Pr(hits ≤ 1 hit in the next hour | average one hit every 12.0 seconds)?

a. 4.04%
b. 6.06%
c. 8.08%
d. 9.09%

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