Question

A beach resort buy a policy to insure against loss of revenues due to major storms in the
summer. The policy pays a total of $ 50,000 if there is only one major storms during
the summer, a total of $ 100,000 if there are two major storms, and a total payment
of $ 200,000 if there are more than two major storms. The number of major storms in
one summer is modeled by a Poisson distribution with mean of 0.5 per summer. Find

(a) Find the expected premium for this policy during one summer.

(b) the standard deviation of the cost of providing this insurance for one summer.

Answer 1

$$P(X=x)={e^{-\lambda}\cdot\lambda^x\over x!}$$

$\lambda=0.5$

$$P(X=1)={e^{-0.5}\cdot(0.5)^1\over 1!}\approx0.30327$$

$$P(X=2)={e^{-0.5}\cdot(0.5)^2\over 2!}\approx0.07582$$

$$P(X>2)=1-P(X=0)-P(X=1)-P(X=2)=$$

$$=1-{e^{-0.5}\cdot(0.5)^0\over 0!}-{e^{-0.5}\cdot(0.5)^1\over 1!}-$$

$$-{e^{-0.5}\cdot(0.5)^2\over 2!}\approx0.01439$$

$$E(X)=50000\cdot0.30327+100000\cdot0.07582+$$

$$+200000\cdot0.01439=25623.50$$

$$E(X^2)=50000^2\cdot0.30327+100000^2\cdot0.07582+$$

$$+200000^2\cdot0.01439=2091975000$$

$$Var(X)=\sigma^2=E(X^2)-(E(X))^2=$$

$$=2091975000-(25623.50)^2=1435411247.75$$

$$\sigma=\sqrt{\sigma^2}=\sqrt{1435411247.75}=37886.82$$