annual premium

I do not understand what the question means...can someone give me some help, please?

The annual premium of a special kind of insurance starts at $1000 and is reduced
by 15% after each year where no claim has been filed. The probability that a claim
is filed in a given year is 0.1, independently of preceding years. What is the PMF of
the total premium paid up to and including the year when the first claim is filed

Is this the answer? I dont really know how to answer this

P((k sigma x=1)1000*(0.75)^(x-1))=0.1*(0.9)^(k-1)

I like this question. It's a good application of Geometric distribution.

Define the random variable Y to be the year in which the first accident happens. Then Y takes values on the set {1,2,3,...}, and Y follows a geometric distribution (# of trials needed for the first success) with success probability p=0.1.

\(\displaystyle \L P(Y=k) = (1-p)^{k-1} p = 0.9^{k-1} 0.1\)

Having that set up, we can define X to be the random variable that gives the sum of the premiums paid until (and including) the year in which the first accident happens. I think we can assume that the premium for a year is paid at the beginning of each year. Then,

\(\displaystyle \L X = \sum_{j=0}^Y 1000 (1-0.15)^j\)
To find the PMF of X, calculate the following probability,

\(\displaystyle \L P(X=A) = P\left( \sum_{j=0}^Y 1000 (0.85)^j = A \right)
= P\left( 1000 \frac{1-0.85^{Y+1}}{1-0.85} =A \right)
= ?\)

You can find this using the first line above.

So I guess my answer was right except from 1-0.15=0.85 not 0.75
and also forgot to put =A inside P()
(also PX(x)=0 otherwise)
Thank you!