Probability Theory - A Mathematical Basis for
Making Decisions under Risk and Uncertianty:
Lecture III
Charles B. Moss
August 24, 2010
I. Introduction
A. In the vernacular of the statistician the unknown or unknowable
event is called a random variable.
1.

The Farm Portfolio Problem: Part I
Lecture XII
I. Deriving the EV Frontier
A. The discussion over the past two weeks has touched on the equivalence between
the mean-variance approach and direct utility maximization. Now, I want to
further develop the EV a

The Farm Portfolio Problem: Part I
Lecture XII
An Empirical Model of MeanVariance
Deriving the EV Frontier
Let us begin with the traditional portfolio model. Assume that we want to minimize the variance associated with attaining a given level of income.

Von Neumann-Morgenstern - Proof I: Lecture
XII
Charles B. Moss
September 16, 2010
I. A:A If u v then < implies
(1 ) u + v (1 ) u + v
(1)
1. The direction of the assertion is that if u v and < , then the
preference ordering must follow.
2. To demonstrate t

Farm Portfolio Problem: Part II Lecture XIII
I. Hazell, P.B.R. A Linear Alternative to Quadratic and Semivariance Programming for Farm Planning Under Uncertainty. American Journal of Agricultural Economics 53(1971):53-62. A. This article is the basis for

Farm Portfolio Problem: Part II
Lecture XIII
MOTAD
x
Hazell, P.B.R. "A Linear Alternative to Quadratic and Semivariance Programming for Farm Planning Under Uncertainty." American Journal of Agricultural Economics 53(1971):53-62.
Fall 2004
Farm Portfolio P

Von Neumann-Morgenstern - Proof II: Lecture
XIII
Charles B. Moss
September 20, 2010
I. Separating Classes
A. There must exist 0 with 0 < 0 < 1 which separates the classes.
1. Thus, 0 will be such that for < 0 the resulting bundle is
in Class I,
2. And if

Farm Portfolio Problem: Part III
Lecture XIV
Target MOTAD
The target MOTAD model is a two-attribute risk and return model.
Return is measured as the sum of the expected return of each activity multiplied by the activity level.
Fall 2004
2
Risk is measu

Closed Form Solutions to Expected Utility:
Lecture XIV
Charles B. Moss
September 21, 2010
I. Closed Form Solutions
A. By the von Neumann and Morgenstern proof we conclude that
decision makers choose those decisions in a way that maximizes
their expected u

Meyers Location Scale: Lecture XV
Charles B. Moss
September 26, 2010
I. Meyers Location-Scale
A. Denition: Two cummulative distributions functions G1 (.) and
G2 (.) are said to dier only by location and scale parameters and
if G1 (x) = G2 ( + x) with > 0.

Empirical Examples of the Central Limit
Theorem: Lecture XVI
Charles B. Moss
October 7, 2010
I. Back to Asymptotic Normality
A. The characteristic function of a random variable X is dened as
X (t) = E eitX = E [cos (tX ) + i sin (tX )]
= E [cos (tX )] + i

Von Neumann-Morgenstern: Lecture XI
Charles B. Moss
September 14, 2010
I. Numerical Stu
A. In the preceding lecture, we found the expected utility of a gamble that paid $150,000 with probability of 0.6 and $50,000 with
probability 0.4. Assuming a r = 0.5,

AEB 6182, Lecture XI Professor Charles Moss
Value of Information Lecture XI
I. Decision Making and Bayesian Probabilities A. Traditionally, Bayesian analysis involves a procedure whereby new information is integrated into a prior distribution to generate

Lecture V: Derivation of the Minimum-Variance Frontier, MinimumVariance Portfolio, and the Efficient Set with a Risk-Free Asset
I.
Derivation of the Minimum-Variance Frontier
A. The minimum-variance frontier is the locus of points that represent the
minim

Savage State-Dependent Expected Utility Lecture IV
I. Savages state dependent expected utility A. Savages subjective expected utility theory takes as the object of choice state dependent outcomes. Security 1
x11 u1 ( x11 )
Security 2
x21 u1 ( x21 )
x12 u2

Savage State-Dependent Expected Utility
Lecture IV
Savage's state dependent expected utility
Savage's subjective expected utility theory takes as the object of choice state dependent outcomes.
Security 1
x11 u u1 ( x11 )
Security 2
x21 u u1 ( x21 )
x12 u

Conditional Probability and Distribution
Functions: Lecture IV
Charles B. Moss
August 27, 2010
I. Conditional Probability and Independence
A. In order to dene the concept of a conditional probability it is
necessary to dene joint and marginal probabilitie

Expected Value and Moments
Charles B. Moss
August 28, 2010
I. Expected Value
A. The random variable can also be described using a statistic.
1. One basic statistic encountered by students in statistics courses
is the mean of a random variable.
2. Denition

Moment Generating Function and Method of
Moments: Lecture VI
Charles B. Moss
August 31, 2010
I. Moment Generating Function
A. Associated with each distribution is a unique function called the
moment generating function that can be used to derive the momen

Maximum Likelihood and Examples: Lecture
VII
Charles B. Moss
September 2, 2010
I. Maximum Likelihood
A. An alternative objective approach to estimating the parameters
of a distribution function is by maximum likelihood.
1. The argument behind maximum like

Empirical Maximum Likelihood: Lecture VIII
Charles B. Moss
September 10, 2010
I. Empirical Maximum Likelihood and Stochastic Process
A. To demonstrate the estimation of the likelihood functions using
maximum likelihood, we formulate the estimation problem

Martingales and Random Walks: Lecture IX
Charles B. Moss
September 10, 2010
I. Martingales
A. Suppose that we have a sequence of random variables Xt : t =
1, 2, (or X1 , X2 , X3 ) dened on a measure space (C ,B ,P ).
1. As with most of our models of risk,

Expected Utility: Lecture X
Charles B. Moss
September 10, 2010
I. Basic Utility
A. A typical economic axiom is that economic agents (consumers,
producers, etc.) behave in a way that maximizes their expected
utility. The typical formulation is
max U (x1 ,

Lecture XVII
Development of FSD and SSD
I.
The Concept of an Efficiency Criteria
A. An efficiency criteria is a decision rule for dividing alternatives into two
mutually exclusive groups: efficient and inefficient.
1. If an alternative is in the efficient

Risk Aversion in the Large and Small: Lecture
XVII
Charles B. Moss
October 4, 2010
I. Basics of Risk Aversion
A. Back to Friedman and Savage:
1. An economic agen with a von Neumann-Morgenstern utility
function v : R R is weakly risk averse if and only if

Lecture XVIII Increasing Risk
I. Literature Required A. Over the next several lectures, I would like to develop the notion of stochastic dominance with respect to a function. Meyer is the primary contributor to the basic literature, so the primary reading

Conditional Probability and Distribution
Functions: Lecture IV
Charles B. Moss
August 27, 2010
I. Conditional Probability and Independence
A. In order to dene the concept of a conditional probability it is
necessary to dene joint and marginal probabilitie

Moment Generating Function and Method of
Moments: Lecture VI
Charles B. Moss
August 31, 2010
I. Moment Generating Function
A. Associated with each distribution is a unique function called the
moment generating function that can be used to derive the momen

Maximum Likelihood and Examples: Lecture
VII
Charles B. Moss
September 2, 2010
I. Maximum Likelihood
A. An alternative objective approach to estimating the parameters
of a distribution function is by maximum likelihood.
1. The argument behind maximum like

Empirical Maximum Likelihood: Lecture VIII
Charles B. Moss
September 10, 2010
I. Empirical Maximum Likelihood and Stochastic Process
A. To demonstrate the estimation of the likelihood functions using
maximum likelihood, we formulate the estimation problem