Discrete Probability Distribution

I have been unable to solve the following exercise from my Probability and Statistics book.

Quote:

Under an insurance policy, a maximum of five claims may be filed per year by a policyholder. Let

be the probability that a policyholder files n claims during a given year, where

= 1,2,3,4,5. An actuary makes the following observations:

(i)

for

= 1,2,3,4.

(ii) The difference between

and

is the same for

= 1,2,3,4.

(iii) Exactly 40% of the policyholders file fewer than three claims during a given year.

Calculate the probability that a random policyholder will file more than three claims during a given year.

(A) 0.14 (B) 0.16 (C) 0.27 (D) 0.29 (E) 0.33

I succeeded in expressing observations (ii) and (iii) as mathematical statements:

(ii) for = 1,2,3,4.

(iii)

I also know the probabilities sum to 1, so

To solve the exercise, I need to find , but I have been unable to do so. I tried rewriting the sum of probabilities like so:

I have tried other approaches, but it seems I am aimlessly solving equations that are just restatements of each other. I believe if I could find and , then I could find and . If anyone could point me in the right direction, I would appreciate it.

By the way, the solutions manual says the answer is (C) 0.27.

Re: Discrete Probability Distribution

Quote:

Originally Posted by

**NOX Andrew** I have been unable to solve the following exercise from my Probability and Statistics book.

I succeeded in expressing observations (ii) and (iii) as mathematical statements:

(ii)

for

= 1,2,3,4.

(iii)

I also know the probabilities sum to 1, so

To solve the exercise, I need to find

, but I have been unable to do so. I tried rewriting the sum of probabilities like so:

I have tried other approaches, but it seems I am aimlessly solving equations that are just restatements of each other. I believe if I could find

and

, then I could find

and

. If anyone could point me in the right direction, I would appreciate it.

By the way, the solutions manual says the answer is (C) 0.27.

i dont understand the proposed solution.

satisfies all the conditions and would give an answer of 0.4.

Moreover, the above solution is unique:

(ii)

=>

(iii)

=>

=> **[1]**

And as you already noticed:

=> **[2]**

The only solutions to [1] and [2] are a=0.2, k=0

and so** **

Re: Discrete Probability Distribution

Thank you for your reply. In **[2]**, it appears you made the substitution , but only applies for . Does this change anything?

Since it doesn't apply to , a possible distribution satisfying the conditions is as follows , , , , , in which case the answer is 0.43.

Re: Discrete Probability Distribution

the relationship applies to n=1,2,3,4,**5.**

You already noted that (n=1,2,3,4)

use n=4, so:

and then use the same reasoning as before.

**Remark**

If you believe that the above isn't valid for then there are infinately many solutions to the problem(this isn't hard to show).

Re: Discrete Probability Distribution

Of course! It seems so obvious now. Thank you for understanding my question about applying it to p_5 and providing such a clear, concise explanation.

Re: Discrete Probability Distribution

FYI there is a mistake in my first post

when i said

I changed the sign of k. This doesn't affect the result