Before I begin, please note: I hated this chapter. If there are any errors please let me know asap!

A deductible is an amount that is subtracted from an insurance claim. If you have a $500 deductible on your car insurance, your insurance company will only pay damages incurred beyond $500. We are interested in the following random variables: and .

**Definitions:**

*Payment per Loss*:*Limited Payment per Loss*:

**Expected Values:**

We may also be interested in the payment per loss, given payment is incurred (

*payment per payment*) .By definition:

Since actuaries like to make things more complicated than they really are, we have special names for this expected value.

*It is denoted by**and is called mean excess loss*in P&C insurance and is called*mean residual life*in life insurance*.*Weishaus simplifies the notation by using the P&C notation without the random variable subscript. I’ll use the same.**Memorize!**

- For an
*exponential distribution*,

- For a
*Pareto*distribution,

- For a
*single parameter Pareto*distribution,

**Useful Relationships:**

**Actuary Speak (important for problem comprehension):**

- The random variable is said to be
*shifted*by and*censored*. - is called
*mean excess loss*or*mean residual life*. - The random variable can be called
*limited expected value*,*payment per loss with claims limit*, and*amount not paid due to deductible*. can be called a*claims limit*or*deductible*depending on how it is used in the problem. - If data is given for with observed values and number of observations or probabilities, the data is called the
*empirical distribution*. Sometimes empirical distributions may be given for a problem, but you are still asked to assume an*parametric distribution*for .