Question

(b) A company prices its hurricane insurance usng the following assumptsons (i) In any calendar year, there can be at most one hurricane (1) In any calendar year, the probability of a hurricane is 0.05 (iii) The mumber of hurricanes in any calendar year is independent of the number of hurricanes in any other calendar year e .os Using the company's assumptions, for a 20-year period i (10 points) what are the expected value and standard deviation of the number of hurricanes?

Answer 1

i) Define X_i=number of hurricanes in i th  year, i=1,2,..,20

Then X_i is either 0 or 1.

The probability p of a hurricane is .05

Y=X₁+...+X₂₀ is the total number of hurricanes in 20 years.

E(Y)=EX₁+...+EX₂₀=20*.05=1 (as E(X_i)=.05)

Var(Y)=Var(X₁)+...+Var(X₂₀) due to independence.

Var(X_i)=E(X_i²)-E²(X_i)=.05-.05*.05=.0475

Var(Y)=20*.0475= 0.95
SD(Y)=square root of Var(Y)= 0.9746794

ii) P(There are fewer than 3 hurricanes)=P(Y<3)=P(Y<=2)= 0.9245163

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