Question

Let X be the damage incurred (in $) in a certain type of accident during a given year. Possible X values are 0, 1000, 5000, and 10000, with probabilities .8, .1, .08, and .02, respectively. A particular company offers a $500 de ductible policy. If the company wishes its expected profit to be $100, what premium amount should it charge?

Answer 1

Answer:

• The premium amount will be define as:        

E(X) = $100 x 0,81 + $600 * 0,1 + $4,600 * 0,08 + $9,600 * 0,02        E(X) = $81 + $60 + $368 + $192 = $701      

Explanation:

We define E(X) as the expected value of the premium amount that we need to calculate.    

The E(X) is a weighted average of the possible values that x can take according to the probability of occurence of X    

The company offers a $500 deductible and expect to get a profit of $100    

Let Y denotes the premium amount:    

If X = 0    

Then Y = X + $100    

Y = $100    

Otherwise    

Y = X - $500 + $100  = X - $400    

If X = $0    

Y = ( $0 + $100 ) * 0,81 = $100 * 0,81    

If X = $1,000    

Y = ( $1.000 - $500 + $100 ) * 0,1 = $600 * 0,1    

If X = $5,000    

Y = ( $5.000 - $500 + $100 ) * 0,08 = $4,600 * 0,08    

If X = $10,000    

Y = ( $10.000 - $500 + $100 ) * 0,02 = $9,600 * 0,02    

The premium amount will be define as:    

E(X) = $100 x 0,81 + $600 * 0,1 + $4,600 * 0,08 + $9,600 * 0,02    

E(X) = $81 + $60 + $368 + $192 = $701