The annual premium of a special kind of insurance starts at

The annual premium of a special kind of insurance starts at $1000 and is reduced by 10% after each year where no claim has been ?led. The probability that a claim is ?led in a given year is 0.05, independently of preceding years. What is the PMF of the total premium paid up to and including the year when the ?rst claim is ?led?

Solution

P((k sigma x=1)1000*(0.90)^(x-1))=0.05*(0.95)^(k-1)