annual premium

[taz]

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

 

[taz]

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)

 

[mark07]

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.

 

[taz]

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!