1. Let X follow a negative binomial distribution with r = 4 and β= 1.2. What is the probability that X = 8?
2. X follows an extended modi ed negative binomial distribution with r =-0.8 and β = 2, and p0 = 0.4. What is P(X = 7)?
3.Let X follow a compound Poisson-Negative binomial distribution with parameters λ= 3.3, r = 4.8 and β= 2.3. Calculate the conditional probability that X = 7 given that X ≤10.
4. Let X follow a mixed negative binomial distribution with β = 1.5 and r following a gamma distribution with α = 2 and θ= 4. What is the probability that X = 2?
5. An insurance company estimates that the number of claims made by an individual in a year follows a Poisson distribution with parameter , where varies between individuals, following a gamma distribution with α= 3 and β= 0.05.
(a) What is the probability that a randomly chosen individual makes 3 claims in a given year?
(b) If an individual has made 3 claims in a given year, what is the proba- bility that that individual makes 3 claims in the next year?