CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 4. Discrete Probability Distribution

Mean Value or Expected Value

The mean value or expected value of a discrete random variablexis given by

μ=E(x) =∑

x

x p(x)

This definition is equivalent to the simpler one you have learned before:

μ=

1

n

n

∑

i= 1

xi

However, the simpler definition would not be usable for many of the probability distributions in statistics.

Example:

An insurance company sells life insurance of $15,000 for a premium of $310 per year. Actuarial tables show that
the probability of death in the year following the purchase of this policy is 0.1%. What is the expected gain for this
type of policy?

Solution:

There are two simple events here, either the customer will live this year or will die. The probability of death, as
given by the problem, is 0.1% and the probability that the customer will live is 1− 0. 001 = 99 .9%. The company’s
expected gain from this policy in the year after the purchase is the random variable, which can have the values shown
in the table below.

TABLE4.3:

Gain,x Simple events Probability

$310 Live 99 .9%

−$14, 690 Die 0 .1%

Figure:Analysis of the possible outcomes of an insurance policy

Remember, if the customer lives, the company gains $310 as a profit. If the customer dies, the company gains
$310−$15, 000 =−$14,690 (a loss). Therefore, the expected profit is,

μ=E(x) =∑

x

x p(x)

μ= ( 310 )( 99 .9%)+( 310 − 15 , 000 )( 0 .1%)

= ( 310 )( 0. 999 )+( 310 − 15 , 000 )( 0. 001 )

= 309. 69 − 14. 69 =$295

μ=$295

This tells us that if the company would sell a very large number of the 1−year $15,000 policy to too many people,
it will make on average a profit of $295 per sale next year.

Another approach is to calculate the expected payout, not the expected gain