Ch 1 Flashcards
What does the parameter λ represent in the context of motor insurance claims?
The average rate of occurrence of claims (e.g., 50 per day).
What distribution does the time to the first claim follow in a Poisson process?
Exponential distribution with parameter λ.
What is the formula for the probability that the time to the first claim is greater than t?
P(T1 > t) = exp{-λt}.
Fill in the blank: The time between claims in a Poisson process has an _______ distribution with parameter λ.
exponential
What property does the exponential distribution have that relates to the time until the next event?
Memoryless property.
If reported claims follow a Poisson process with rate 5 per day, what is the probability of fewer than 2 claims reported on a given day?
0.040.
What is the expected number of claims reported on a given day if the rate is 5 per day?
5.
What is the probability that another claim will be reported during the next hour if the rate is 5 claims per day?
0.1881.
What assumptions are made for the compound Poisson process regarding the random variables {Xi}?
- Independent and identically distributed
- Independent of N(t) for all t ≥ 0.
What is the formula for the moment generating function of the random variable Xi?
M_X(r) = E[e^{rX}].
What is the mean of the aggregate claims process S(t) in a compound Poisson process?
λtm1.
What is the variance of the aggregate claims process S(t) in a compound Poisson process?
λtm2.
What is the intuitive reason for the assumption c > λm1 in the context of premium income?
To ensure the insurer charges premiums greater than expected claims outgo.
What is the distribution of the claims arising from an annual insurance policy?
Normal distribution with mean 0.7P and standard deviation 2.0P.
What is the formula for the insurer’s surplus at the end of the coming year?
U(1) = initial surplus + premiums - expenses - claims.
What is the probability that the insurer will prove to be insolvent at the end of the coming year?
0.067.
How is the surplus at the end of the second year calculated?
U(2) = initial surplus + premiums - expenses - claims.
What is the distribution of S(2) if the insurer expects to sell 200 policies?
N(1.05 m, (0.173 m)²).
What is the probability that the surplus will be negative at the end of the second year?
To be calculated based on distribution of S(2).
What is the probability that the surplus will be negative at the end of the second year?
0.074
This is calculated as (P[U(2)<0] = P[S(2)>1.3 mathrm{~m}])
What distribution is assumed for the number of claims from a portfolio of policies?
Poisson distribution with parameter 30 per year
What is the individual claim amount distribution in this scenario?
Lognormal with parameters ( mu=3 ) and ( sigma^{2}=1.1 )
What is the rate of premium income from the portfolio?
1,200 per year
What is the expected number of claims in a two-year period?
60
What is the mean of aggregate claims in a two-year period?
2,088.80
What is the variance of aggregate claims in a two-year period?
218,457
When does ruin occur for an insurer?
When (S(2) > ext{initial surplus} + ext{premiums received})
What is the probability of ruin calculated in the example?
0.0025
What is the percentage of security loading typically represented by?
0.2 or 20%
What is the formula for the rate of premium income when related to claims outgo?
c=(1+ heta) lambda m_{1}
What are the mean and variance formulas for total claim amount in a compound Poisson process?
- Mean: (E[S(t)]=lambda t E[X])
- Variance: (operatorname{Var}(S(t))=lambda t E[X^{2}])
What does the moment generating function of the process represent?
(M_{S(t)}(r)=exp left(lambda tleft(M_{X}(r)-1
ight)
ight))
What is the condition for (M_{X}(r)) to be finite?
There exists some number (gamma) such that (M_{X}(r)) is finite for all (r<gamma)
What is the limit of the moment generating function as (r) approaches (gamma)?
(lim {r
ightarrow gamma^{-}} M{X}(r)=infty)
What is the adjustment coefficient used for?
It is associated with the surplus process and helps determine the probability of ruin.
What does Lundberg’s inequality state?
(psi(U) leq exp {-R U})
What is (psi(U)) in the context of Lundberg’s inequality?
The probability of ultimate ruin
