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Expected Value- Random variables. Def. A random variable , X , is a numerical measure of the outcomes of an experiment. Example:. Experiment- Two cards randomly selected Let X be the number of diamonds selected. Events can be described in terms of random variables Example:

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expected value random variables

Expected Value- Random variables

Def. A randomvariable, X, is a numerical measure of the outcomes of an experiment

example
Example:
  • Experiment- Two cards randomly selected

Let X be the number of diamonds selected

slide3
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
slide6
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

slide8
Def. The mean (or expectedvalue) of X gives the value that we would expect to observe on average in a large number of repetitions of the experiment
important
Important
  • Concept of Expected value describe the expected monetary return of experiment

Sum of the values, weighted by their respected probabilities

slide10
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?

tools to calculate e x project a
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 x1 ,x2 ,x3

X= x1 is the event that we have Loss

X= x2 is the event that we are breaking even

X= x3 is the event that we have a Profit

  • x1=$-26,000
  • x2=$0
  • x3=$68,000
  • P(X= x1)=0.3
  • P(X= x2)= 0.5
  • P(X= x3)= 0.2
tools to calculate e x project b
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 x1 ,x2 ,x3

X= x1 is the event that we have Loss

X= x2 is the event that we are breaking even

X= x3 is the event that we have a Profit

  • x1=$-71,000
  • x2=$0
  • x3=$143,000
  • P(X= x1)=0.2
  • P(X= x2)= 0.65
  • P(X= x3)= 0.15
focus on the project
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

tools to calculate e x
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

x1=$ 4,000,000

x2=$ 250,000

slide16

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

  • Using Expected value formula
decision
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

slide18
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.