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Expected Value- Random variables Def. A random variable, X, is a numerical measure of the outcomes of an experiment
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Example: Experiment- Two cards randomly selected Let X be the number of diamonds selected
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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
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Probabilities of events can be stated as probabilities of the corresponding values of X
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Example:
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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
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Suppose that X can only assume the values x 1, x 2,... x n. Then
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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
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Important Concept of Expected value describe the expected monetary return of experiment Sum of the values, weighted by their respected probabilities
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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 $143,000 with probability 0.15. Which investment is better?
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Tools to calculate E(X)-Project A Random Variable (X)- The amount of money received from the investment in Project A X can assume only x 1, x 2, x 3 X= x 1 is the event that we have Loss X= x 2 is the event that we are breaking even X= x 3 is the event that we have a Profit x 1 =$-26,000 x 2 =$0 x 3 =$68,000 P(X= x 1 )=0.3 P(X= x 2 )= 0.5 P(X= x 3 )= 0.2
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Tools to calculate E(X)-Project B Random Variable (X)- The amount of money received from the investment in Project B X can assume only x 1, x 2, x 3 X= x 1 is the event that we have Loss X= x 2 is the event that we are breaking even X= x 3 is the event that we have a Profit x 1 =$-71,000 x 2 =$0 x 3 =$143,000 P(X= x 1 )=0.2 P(X= x 2 )= 0.65 P(X= x 3 )= 0.15
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Focus on the Project How can Expected value help us with the decision on whether or not to attempt a loan workout? Recall: Events S- An attempted workout is a Success F- An attempted workout is a Failure
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Tools to calculate E(X) Random Variable (X)- The amount of money Acadia receives from a future loan workout attempt X can assume only Full Value Default Value x 1 =$ 4,000,000 x 2 =$ 250,000
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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.
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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
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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.
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