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Finding the Expected Value

 

Example:  A term life insurance policy will pay a beneficiary a certain sum of money upon the death of the policyholder.  These policies have premiums that must be paid annually.  Suppose a life insurance company sells a $250,000 one-year term life insurance policy to a 20-year-old male for $350.  According to the National Vital Statistics Report, Vol. 47,

No. 28, the probability the male will survive the year is 0.99865.  Compute the expected value of this policy to the insurance company.

 

Step 1: We have P(survives) = 0.99865, so the P(dies) = 0.00135.  From the point of view of the insurance company, if the client survives for the year, the insurance company makes $350.  Therefore, we let x = $350 if the client survives the year.  If the client dies during the year, the insurance company must pay $250,000 to the client’s beneficiary; however, the company still keeps the $350, so we let  The value is negative because it is money paid out by the insurance company.  We have the probability distribution listed below.

 

$350(survives)

0.99865

-$249,650(dies)

0.00135

 

Step 2:  Substituting into Formula (3), we obtain the expected value (from the point of view of the insurance company) of the policy.

 

 

Interpretation:  The company expects to make $12.50 for each 20-year-old male client they insure.

 

The $12.50 profit of the insurance company is a long-term result.  Of course, it does not make $12.50 on each person it insures, but rather the average profit per person insured is $12.50.

 

Press STAT and select 1: Edit. Clear L1 and L2. Enter the X-values from the above table into L1 and the associated probabilities into L2.

 

 

Notice that the value of -249,650 appears as -2.5E5. This is a rounded value and it is written in scientific notation. The actual value is stored in the calculator; the rounded value is for display purposes only.

 

Press STAT and highlight CALC. Select 1:1 Var stats, press ENTER and press 2nd [L1] , 2nd [L2] ENTER to see the descriptive statistics.

 

 

The expected value of this discrete random variable is 12.5.