1. Expected Value- Random variables Def. A random variable, X, is a numerical measure of the outcomes of an experiment

2. Example:  Experiment- Two cards randomly selected Let X be the number of diamonds selected

3.  Events can be described in terms of random variables Example:  is the event that exactly one diamond is selected  is the event that at most one diamond is selected

4. Probabilities of events can be stated as probabilities of the corresponding values of X

5.  Example:

6. In general, is the probability that X takes on the value x is the probability that X takes on a value that is less than or equal to x

7. Suppose that X can only assume the values x 1, x 2,... x n. Then

8.  Def. The mean (or expected value) of X gives the value that we would expect to observe on average in a large number of repetitions of the experiment

9. Important  Concept of Expected value describe the expected monetary return of experiment Sum of the values, weighted by their respected probabilities

10.  Example (Exercise 13): An investment in Project A will result in a loss of $26,000 with probability 0.30, break even with probability 0.50, or result in a profit of $68,000 with probability 0.20. An investment in Project B will result in a loss of $71,000 with probability 0.20, break even with probability 0.65, or result in a profit of $143,000 with probability 0.15. Which investment is better?

11. Tools to calculate E(X)-Project A  Random Variable (X)- The amount of money received from the investment in Project A  X can assume only x 1, x 2, x 3 X= x 1 is the event that we have Loss X= x 2 is the event that we are breaking even X= x 3 is the event that we have a Profit  x 1 =$-26,000  x 2 =$0  x 3 =$68,000  P(X= x 1 )=0.3  P(X= x 2 )= 0.5  P(X= x 3 )= 0.2

12. Tools to calculate E(X)-Project B  Random Variable (X)- The amount of money received from the investment in Project B  X can assume only x 1, x 2, x 3 X= x 1 is the event that we have Loss X= x 2 is the event that we are breaking even X= x 3 is the event that we have a Profit  x 1 =$-71,000  x 2 =$0  x 3 =$143,000  P(X= x 1 )=0.2  P(X= x 2 )= 0.65  P(X= x 3 )= 0.15

14. Focus on the Project  How can Expected value help us with the decision on whether or not to attempt a loan workout?  Recall: Events S- An attempted workout is a Success F- An attempted workout is a Failure

15. Tools to calculate E(X)  Random Variable (X)- The amount of money Acadia receives from a future loan workout attempt  X can assume only Full Value Default Value x 1 =$ 4,000,000 x 2 =$ 250,000

16.  Using Expected value formula The sheet Expected Value in the Excel file Loan Focus.xls performs the following computation for the expected value of X.

17. Decision? Recall  Bank Forecloses a loan if Benefits of Foreclosure > Benefits of Workout  Bank enters a Loan Workout if Expected Value Workout > Expected Value Foreclose

18. Since the expected value of a work out is $1,991,000 and the “expected value” of foreclosing is a guaranteed $2,100,000, it might seem that Acadia Bank should foreclose on John Sanders’ loan.