Expected Value- Random variables
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Transcript Expected Value- Random variables
Expected ValueRandom 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
CC
DC
S
HC
SC
CD
DD
HD
CH
DH
HH
SD
SH
CS
DS
HS
SS
Events can be described in terms of
random variables
Example:
X 1
is the event that exactly one
diamond is selected
X 1
is the event that at most one
diamond is selected
Probabilities of events can be stated
as probabilities of the corresponding
values of X
P( X 1) P( F )
6
16
3
8
Example:
CC
P( X 1) P DH
HS
15
16
CD CH CS DC
DS HC HD HH
SC SD SH SS
In general,
is the probability that X
takes on the value x
P( X x)
is the probability that X
takes on a value that is less than or
equal to x
P( X x)
Suppose that X can only assume the
values x1, x2, ... xn. Then
n
P( X x ) 1
i 1
i
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
n
X E ( X ) xi * P( X xi )
i 1
Important
Concept of Expected value describe
the expected monetary return of
experiment
n
X E ( X ) xi * P( X xi )
i 1
Sum of the values,
weighted by their
respected probabilities
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
$143,000 with probability 0.15. Which
investment is better?
Tools to calculate E(X)-Project A
Random Variable (X)- The amount of
money received from the investment in
Project A
X can assume only x1 ,
X= x1 is the event that we
X= x2 is the event that we
X= x3 is the event that we
x1=$-26,000
x2=$0
x3=$68,000
P(X= x1)=0.3
P(X= x2)= 0.5
P(X= x3)= 0.2
x2 , x3
have Loss
are breaking even
have a Profit
Tools to calculate E(X)-Project B
Random Variable (X)- The amount of
money received from the investment in
Project B
X can assume only x1 ,
X= x1 is the event that we
X= x2 is the event that we
X= x3 is the event that we
x1=$-71,000
x2=$0
x3=$143,000
P(X= x1)=0.2
P(X= x2)= 0.65
P(X= x3)= 0.15
x2 , x3
have Loss
are breaking even
have a Profit
Project A :
E ( X ) 0.30 ($26,000) 0.50 $0 0.20 $68,000
$5800
Project B :
E ( X ) 0.20 ($71,000) 0.65 $0 0.15 $143,000
$7250
Focus on the Project
How can Expected value help us
with the decision on whether or not
to attempt a loan workout?
Recall:
Events
S- An attempted workout is a Success
F- An attempted workout is a Failure
Tools to calculate E(X)
Random Variable (X)- The amount
of money Acadia receives from a
future loan workout attempt
X can assume only
Full Value
Default Value
x1=$ 4,000,000
x2=$ 250,000
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.
E ( X ) $4,000,000 P ( X $4,000,000) $250,000 P ( X $250,000)
$4,000,000 P ( S ) $250,000 P ( F )
$4,000,000 (0.464) $250,000 (0.536)
$1,991,000
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
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.