Todays Goals

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Todays Goals

• Introduction to Probability
• Random Variables
• Expected Values
• Risk
• Homework 2 (due Thursday September 25)
• GPCs New Product Decision
• For 1, just draw the ID
• For 2, use tools provided with book.
• Strenlar
• Screening for Colorectal Cancer (Ch. 7)

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Bayes Theorem
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Random Variable

• A Random Variable is a Real-valued function over
  a sample space.
• Let x 1 if you get at least one head in 5 flips,
  0 otherwise.
• Sample space is HHHHH HHHHT HHHTH
• Let x the distance from Amherst to Atlanta, GA
  Sample space is the positive real numbers.

4
Random Variable

• A random variable Y is said to be discrete if it
  can assume only a finite or countably finite
  number of distinct values.
• Discrete Let x 1 if you get at least one head
  in 5 flips, 0 otherwise.
• Sample space is HHHHH HHHHT HHHTH
• Continuous Let x the distance from Amherst to
  Atlanta, GA
• Sample space is the positive real numbers.

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Probability distribution of a discret RV

• The probability that Y takes on the value of y,
  P(Y y), is defined as the sum of the
  probabilities of all sample points in S that are
  assigned the value y
• Notation P(Y y) p(y)
• The probability distribution for a discrete
  random variable Y can be represented by a
  formula, a table, or a graph, which provides the
  probabilities p(y) corresponding to each and
  every value of y.

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Probability Mass Function
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Cumulative Distribution Function

• The CDF gives the probability that a random
  variable is less than or equal to a value.
• The CDF is continuous even when the RV is
  discrete.

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Cumulative Distribution Function
P(Yy)
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Continuous Distributions

• The uncertain quantity can take on any value
  within a range.
• The probability of equaling any particular value
  is zero, since there are infinitely many values
  it can equal.
• We can still construct a CDF P(Yy) F(y)

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Probability Density Function

• Analogous to the PMF is a Probability Density
  Function, f(.)

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Example

• Let f(x) 2-2x for x between 0 and 1.
• Is this a pdf?
• What is P(X ½)?
• What is P(X lt ½)?
• What is P(X ½)?
• What is F(x)?

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Bayes Rule for continuous distributions

• L(yx) is the likelihood function for getting
  signal y given that the value of Xx
• p(x) prior probability density of X
• P(xy) is the posterior probability density over
  X, after getting signal y.

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Moments

• Expected Value (first moment)
• Variance and Standard Deviation (based on the 2nd
  moment)
• Covariance and Correlation

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Expected Value
Only if g is linear is it true that
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The Flaw of Averages

• Jensens inequality
• If g is convex then
• What if g is concave?
• What if g is linear?
• What if g is neither convex nor concave?

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The Flaw of Averages

• Let X 1, 2, 3 with equal likelihood
• What is E1/x?

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A sobering example of the Flaw of Averages (taken
from Sam Savages Insight.xla)
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Flaw of Averages Example

• You are managing a project.
• There are 8 simultaneous tasks that need to get
  done. The project is done when all the tasks are
  done.
• The mean and median time for each task to get
  done is 3 months.

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Flaw of Averages Example

• You are managing a project.
• There are 8 simultaneous tasks that need to get
  done. The project is done when all the tasks are
  done.
• The mean and median time for each task to get
  done is 3 months.
• How long will it take you to finish the project,
  on average?

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Flaw of Averages Example

• You are managing a project.
• There are 8 simultaneous tasks that need to get
  done. The project is done when all the tasks are
  done.
• The mean and median time for each task to get
  done is 3 months.
• What is the probability that you will finish the
  job in about 3 months?

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Flaw of averages Example

• See simulation

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Variance and Standard Deviation

• var(X) EX-EX EX2 EX2
• SD(X)
• Var(aXb) a2 Var(X)
• Co-Variance E(X-EX)(Y-EY)
  EXY-EXEY
• Correlation rxy

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Correlation and Independence

• If two variables have zero correlation, are they
  independent?
• If two variables are independent, do they have
  zero correlation?

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Correlation

• What is correlation?
• Does correlation imply relevance/dependence?
• Does lack of correlation imply independence?

y
x
Are x and y correlated? Are they independent?
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Example

• Find the correlation between X and Y.
• Are they independent? Prove it.

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Variance and Risk

• Variance is a measure of risk
• but it is not the only thing that matters.
• Deal A earn 1,000,000 with probability 0.1 or
  0 with probability 0.9
• Deal B earn 200,000 with probability 0.9 or
  lose 800,000 with probability 0.1
• Which would you prefer? What is the expected
  value and variance of A and of B?

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Risk Profiles and Dominance

• We can consider the risk profiles of different
  strategies.
• Sometimes, one strategy will dominate another
  The probability of getting a payoff of x or
  better is higher, for every x.

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Decision Tree
To build a risk profile of a strategy, collapse
the tree down only to that strategy
10-.25-1 8.75M
Continue
P.1
-1-.25 -1.25M
Invest in first stage
-.25M
Stop
Continue
-1-.25M
1-P.9
-.25M
Stop
0
Dont Invest in first stage
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Decision Tree
To build a risk profile of a strategy, collapse
the tree down only to that strategy
10-.25-1 8.75M
Continue
P.1
-1-.25 -1.25M
Invest in first stage
1-P.9
-.25M
Stop
0
Dont Invest in first stage
30
Decision Tree
To build a risk profile of a strategy, collapse
the tree down only to that strategy then
collapse the probabilities
10-.25-1 8.75M
P.1
Invest in first stage
-1-.25 -1.25M
1-P.9
-.25M
0
Dont Invest in first stage
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Decision Tree
Now we can draw a CDF of invest strategy.
8.75M
.05
Invest in first stage
.05
-1.25M
0.9
-.25M
0
Dont Invest in first stage
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Is there dominance?
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Is there dominance?
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F dominates G
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Stochastic dominance

• Why is it stochastic dominance?

If x SD y, then the p(xgtk) p(ygtk) for all
k. But, it is possible that y could turn out to
be better than x. (example of 1 for 1-5 versus
1 for 6)
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• Roll back the tree to see whether the firms
  should refuse or accept the 3B offer.
• Now, draw the CDFs for the two law firms.
• Which firm is better?
• Will it depend on risk aversion?