Suppose you are insuring 7000 policy holders. Your task is to model the total loss with Collective Risk Model, and calcu

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Suppose you are insuring 7000 policy holders. Your task is to
model the total loss
with Collective Risk Model, and calculate individual premiums. Use
NegBin(r =
4, β = 17.5) for the frequency distribution. Assume ground-up loss
distribution is
X ∼ Exp(2000).
a) What is the mean and variance for the claim frequency, N ?
b) For ground-up loss X, what is expectation, variance, and second
moment
(E(X), V (X), E(X2))?
c) If this NegBin(r = 4, β = 17.5) disribution was a result of
Poisson-Gamma mix-
ture, how are we modeling claims from each driver and driver
population as a
whole? Brifly explain in few sentences.
d) Now we are considering a deductible of d. What is the expected
total payment
E(S)? Express in terms of d.
e) What is the standard deviation of the total payment SD(S) after
the deductible?
Express in terms of d.
f) Let β, the probability of company not suffering a loss, to be
.9. Express the
loading factor θ in terms of d.
g) Find value of d that will make individual premiums equal to
$10.