# normal approximation to gamma

• Aug 4th 2010, 07:17 AM
i_zz_y_ill
normal approximation to gamma
The owner of a car-rental business wants to insure his fleet of limousines against large losses. The annual damage to the fleet is modelled by the random variable X, which is thought to follow a Gamma(10,2) distribution. The insurance company proposes a policy , under which for damage X, the business will get Y back from the insurer. The random variable Y is defined as Y=g(X), where g is the function given by g(x)=max{x-5,0}
a) Using a normal approximation to the distribution of X, determine the probability that the business will actually make a claim in a given year; that is, calculate P(y>0)

the is my answer but im not entirely sure it is correct: X dist Normall$\displaystyle (\frac{10}{2},\frac{10}{2^2})$
$\displaystyle p(y>0)=p(x-5>0)=p(z>\frac{5-\frac{10}{2}}{\sqrt{\frac{10}{2^2}}})$=0.5

b) Let the i.i.d random variable [tex]X_{1},,,X_{5} represent the annual damage that the limousine fleet suffers over 5 consecutive years. Making use of the same approximation as in part a), calculate the probability that the greatest of these annual damage amaounts is large enough to trigger an insurance claim.

part b) i am unsure of what to do,,i really need to know this method,,any ideas whatsoever
• Aug 5th 2010, 09:21 AM
i_zz_y_ill
think i have solved it
$\displaystyle 5X dist expo(\frac{2}{5})$ this follows through as before giving $\displaystyle dist normal(\frac{10}{\frac{2}{5}},\frac{10}{(\frac{2}{ 5})^2}$ can somenoe confirm for me this is correct?