# Notes: April 21, 2014

Statistics and Probability Chapter 16 Guided Notes Name ________________________________ Date _________________ Period __________

## RANDOM VARIABLES and EXPECTED VALUE

**Read the Introduction on page 342 and the blue box “What is an Actuary?” on page 343.**

### RANDOM VARIABLES

 ________________

Variable

- _________________ value based upon the outcome of a ____________  event. ______________ letters, like _____, are used to denote random variables. Any particular value, such as 5, that a random variable can have will be denoted by a ______________________ letter. _________________

Model

– the collection of all the possible _________ and the _________________ that they occur for the random variable. o

Example A

: Suppose that the death rate in any one year for an insurance company is 1 out of every 1000 people that another 2 out of 1000 suffer some kind of disability and everybody else didn’t require the use of either type of insurance. The probability model for this scenario can be displayed in a table.

Policyholder Outcome

X = Payout

Probability P(X)

Death

x =

Disability

x =

### EXPECTED VALUE

__________________ _____________ -

the theoretical long-run average value of a random variable. It is the _____________ of the probability model. The expected value is denoted by ______ or ________. It is found by __________________ the ______________ of variable values and __________________.

### 𝝁 = 𝑬(𝑿) = ∑(𝒙 ∙ 𝑷(𝒙))

o

Example B:

On average, what can the insurance company in the problem above expected to payout per policy? Is the company making a profit if they charge only \$50 per year for this benefit?

o o

Example B, Part 2:

The expected value can be calculated second way by multiplying each payout by the probability that it occurs!

Explain what this value means in context! Example C:

Someone had to take his minivan in for repair because the air-conditioner was cutting out intermittently. The mechanic identified the problem as dirt in a control unit. The mechanic said that in about 75% of such cases, drawing down and then recharging the coolant a couple of times cleans up the problem – and costs only \$60. If that fails, then the control unit must be replaced at an additional \$100 for parts and \$40 for labor.

 Define the random variable and construct the probability model.

  What is the expected value of the cost of this repair?

What does this value mean in this context?