The purpose of this descriptive case study analysis was to provide portraits of the heuristics students used and difficulties they encountered solving conditional probability problems prior to and after two-week instruction on sample space, probability, and conditional probability. Further analysis consisted of evaluating the data in relation to a previously designed Conditional Probability Framework for assessing students levels of thinking developed by Tarr and Jones (1997). Five volunteer participants from a contemporary college mathematics course participated in pre-and post-interviews of a Probability Knowledge Inventory. The Inventory consisted of seven tasks on sample space, probability, and conditional probability. The semi-structured interviews provided participants' explanations on the development of their solutions to the seven tasks. Among the five participants, rationalizing, finding the odds, computing the percentages, and stating the ratio of a problem were the preferred heuristics used to solve the problems on the Probability Knowledge Inventory. After the two-week instruction, two of the four participants who did not previously use computation of probability to solve the problem changed their use of heuristics. The difficulties the students encountered prior to instruction included understanding the problem; recognizing the original sample space and when it changes; lacking probability vocabulary knowledge; comparing probability after the sample space changed; understanding the difference between probability and odds; and interchanging ratio, odds, and percentages-sometimes incorrectly-to justify their solution. After the two-week instruction, the students' difficulties diminished. Some improvements included a greater ability to understand the question of interest, to recognize the change in the sample space after a conditioning event, to use probability terminology consistently, and to compare probability after the sample space has changed. Comparisons to the Probability Framework revealed that four of the five participants exemplified Level 3 thinking-being aware of the role that quantities play in forming conditional probability judgements. One participant exemplified a Level 4 thinking-being aware of the composition of the sample space, recognizing its importance in determining conditional probability and assigning numerical probabilities spontaneously and with explanation. / Graduation date: 2001

In this thesis, first we replace the condition X ¡Ø y in Huang and Su (2000) by X ¡Ù y and give necessary and sufficient conditions such that there exists a random variable X satisfying that E(g(X)| X ¡Ø y)=h(y) f(y )/ F(y), " y Î CX, where CX is the support of X.Next, we investigate necessary and sufficient conditions such that h(y)=E(g(X) | X ¡Ø y ), for a given function h and extend these results to bivariate case.

Thesis (Ph. D.)--University of California, San Diego, 2004. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of California, San Diego, 2007. / Title from first page of PDF file (viewed June 21, 2007). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references.

This study revolves around operators defined by conditional expectations and operators generated by random measures. Studies of operators in function spaces defined by conditional expectations first appeared in the mid 1950's by S-T.C. Moy [22] and S. Sidak [26]. N. Kalton studied them in the setting of Lp-spaces 0 < p < 1 in [15, 131 and in L1-spaces, [14], while W. Arveson [5] studied them in L2-spaces. Their averaging properties were studied by P.G. Dodds and C.B. Huijsmans and B. de Pagter in [7] and C.B. Huijsmans and B. de Pagter in [lo]. A. Lambert [17] studied their relationship with multiplication operators in C*-modules. It was shown by J.J. Grobler and B. de Pagter [8] that partial integral operators that were studied A.S. Kalitvin et a1 in [2, 4, 3, 11, 121 and the special cases of kernel operators that were, inter alia, studied by A.R. Schep in [25] were special cases of conditional expectation operators. On the other hand, operators generated by random measures or pseudo-integral operators were studied by A. Sourour [28, 271 and L.W. Weis [29,30], building on the studies of W. Arveson [5] and N. Kalton [14, 151, in the late 1970's and early 1980's. In this thesis we extend the work of J.J. Grobler and B. de Pagter [8] on Multiplication Conditional Expectation-representable (MCE-representable) operators. We also generalize the result of A. Sourour [27] and show that order continuous linear maps between ideals of almost everywhere finite measurable functions on u-finite measure spaces are MCE-representable. This fact enables us to easily deduce that sums and compositions of MCE-representable operators are again MCE-representable operators. We also show that operators generated by random measures are MCE-representable. The first chapter gathers the definitions and introduces notions and concepts that are used throughout. In particular, we introduce Riesz spaces and operators therein, Riesz and Boolean homomorphisms, conditional expectation operators, kernel and absolute T-kernel operators. In Chapter 2 we look at MCE-operators where we give a definition different from that given by J.J. Grobler and B. de Pagter in [8], but which we show to be equivalent. Chapter 3 involves random measures and operators generated by random measures. We solve the problem (positively) that was posed by A. Sourour in [28] about the relationship of the lattice properties of operators generated by random measures and the lattice properties of their generating random measures. We show that the total variation of a random signed measure representing an order bounded operator T, it being the difference of two random measures, is again a random measure and represents ITI. We also show that the set of all operators generated by a random measure is a band in the Riesz space of all order bounded operators. In Chapter 4 we investigate the relationship between operators generated by random measures and MCE-representable operators. It was shown by A. Sourour in [28, 271 that every order bounded order continuous linear operator acting between ideals of almost everywhere measurable functions is generated by a random measure, provided that the measure spaces involved are standard measure spaces. We prove an analogue of this theorem for the general case where the underlying measure spaces are a-finite. We also, in this general setting, prove that every order continuous linear operator is MCE-representable. This rather surprising result enables us to easily show that sums, products and compositions of MCE-representable operator are again MCE-representable. Key words: Riesz spaces, conditional expectations, multiplication conditional expectation-representable operators, random measures. / Thesis (Ph.D. (Mathematics))--North-West University, Potchefstroom Campus, 2004.

This study revolves around operators defined by conditional expectations and operators generated by random measures. Studies of operators in function spaces defined by conditional expectations first appeared in the mid 1950's by S-T.C. Moy [22] and S. Sidak [26]. N. Kalton studied them in the setting of Lp-spaces 0 < p < 1 in [15, 131 and in L1-spaces, [14], while W. Arveson [5] studied them in L2-spaces. Their averaging properties were studied by P.G. Dodds and C.B. Huijsmans and B. de Pagter in [7] and C.B. Huijsmans and B. de Pagter in [lo]. A. Lambert [17] studied their relationship with multiplication operators in C*-modules. It was shown by J.J. Grobler and B. de Pagter [8] that partial integral operators that were studied A.S. Kalitvin et a1 in [2, 4, 3, 11, 121 and the special cases of kernel operators that were, inter alia, studied by A.R. Schep in [25] were special cases of conditional expectation operators. On the other hand, operators generated by random measures or pseudo-integral operators were studied by A. Sourour [28, 271 and L.W. Weis [29,30], building on the studies of W. Arveson [5] and N. Kalton [14, 151, in the late 1970's and early 1980's. In this thesis we extend the work of J.J. Grobler and B. de Pagter [8] on Multiplication Conditional Expectation-representable (MCE-representable) operators. We also generalize the result of A. Sourour [27] and show that order continuous linear maps between ideals of almost everywhere finite measurable functions on u-finite measure spaces are MCE-representable. This fact enables us to easily deduce that sums and compositions of MCE-representable operators are again MCE-representable operators. We also show that operators generated by random measures are MCE-representable. The first chapter gathers the definitions and introduces notions and concepts that are used throughout. In particular, we introduce Riesz spaces and operators therein, Riesz and Boolean homomorphisms, conditional expectation operators, kernel and absolute T-kernel operators. In Chapter 2 we look at MCE-operators where we give a definition different from that given by J.J. Grobler and B. de Pagter in [8], but which we show to be equivalent. Chapter 3 involves random measures and operators generated by random measures. We solve the problem (positively) that was posed by A. Sourour in [28] about the relationship of the lattice properties of operators generated by random measures and the lattice properties of their generating random measures. We show that the total variation of a random signed measure representing an order bounded operator T, it being the difference of two random measures, is again a random measure and represents ITI. We also show that the set of all operators generated by a random measure is a band in the Riesz space of all order bounded operators. In Chapter 4 we investigate the relationship between operators generated by random measures and MCE-representable operators. It was shown by A. Sourour in [28, 271 that every order bounded order continuous linear operator acting between ideals of almost everywhere measurable functions is generated by a random measure, provided that the measure spaces involved are standard measure spaces. We prove an analogue of this theorem for the general case where the underlying measure spaces are a-finite. We also, in this general setting, prove that every order continuous linear operator is MCE-representable. This rather surprising result enables us to easily show that sums, products and compositions of MCE-representable operator are again MCE-representable. Key words: Riesz spaces, conditional expectations, multiplication conditional expectation-representable operators, random measures. / Thesis (Ph.D. (Mathematics))--North-West University, Potchefstroom Campus, 2004.

Obtaining the distribution of the profit and loss (PL) of a portfolio is a key problem in market risk measurement. However, existing methods, such as those based on the Normal distribution, and historical simulation methods, which use empirical distribution of risk factors, face difficulties in dealing with at least one of the following three problems: describing the distributional properties of risk factors appropriately (description problem); deriving distributions of risk factors with time horizon longer than one day (time aggregation problem); and deriving the distribution of the PL given the distributional properties of the risk factors (risk aggregation problem). Here, we show that expansion methods can provide reasonable solutions to all three problems. Expansion methods approximate a probability density function by a sum of orthogonal polynomials multiplied by an associated weight function. One of the most important advantages of expansion methods is that they only require moments of the target distribution up to some order to obtain an approximation. Therefore they have the potential to be applied in a wide range of situations, including in attempts to solve the three problems listed above. On the other hand, it is also known that expansions lack robustness: they often exhibit unignorable negative density and their approximation quality can be extremely poor. This limits applications of expansion methods in existing studies. In this thesis, we firstly develop techniques to provide robustness, with which expansion methods result in a practical approximation quality in a wider range of examples than investigated to date. Specifically, we investigate three techniques: standardisation, use of Laguerre expansion and optimisation. Standardisation applies expansion methods to a variable which is transformed so that its first and second moments are the same as those of the weight function. Use of Laguerre expansions applies those expansions to a risk factor so that heavy tails can be captured better. Optimisation considers expansions with coefficients of polynomials optimised so that the difference between the approximation and the target distribution is minimised with respect to mean integrated squared error. We show, by numerical examples using data sets of stock index returns and log differences of implied volatility, and GARCH models, that expansions with our techniques are more robust than conventional expansion methods. As such, marginal distributions of risk factors can be approximated by expansion methods. This solves a part of the description problem: the information on the marginal distributions of risk factors can be summarised by their moments. Then we show that the dependence structure among risk factors can be summarised in terms of their cross-moments. This solves the other part of the description problem. We also use the fact that moments of risk factors can be aggregated using their moments and cross-moments, to show that expansion methods can be applied to both the time and risk aggregation problems. Furthermore, we introduce expansion methods for multivariate distributions, which can also be used to approximate conditional expectations and copula densities by rational functions.

Thesis (Ph. D.)--University of California, San Diego, 2009. / Title from first page of PDF file (viewed August 11, 2009). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 134-136).

Thesis (M.Sc. (Computer Science))--North-West University, Potchefstroom Campus, 2008.

Rational expectations provide people or economic agents making future decision with available information and past experiences. The first approach to the idea of rational expectations was given approximately fifty years ago by John F. Muth. Many models in economics have been studied using the rational expectations idea. The most familiar one among them is the rational expectations version of the Cagans hyperination model where the expectation for tomorrow is formed using all the information available today. This model was reinterpreted by Thomas J. Sargent and Neil Wallace in 1973. After that time, many solution techniques were suggested to solve the Cagan type rational expectations (CTRE) model. Some economists such as Muth [13], Taylor [26] and Shiller [27] consider the solutions admitting an infinite moving-average representation. Blanchard and Kahn [28] find solutions by using a recursive procedure. A general characterization of the solution was obtained using the martingale approach by Broze, Gourieroux and Szafarz in [22], [23]. We choose to study martingale solution of CTRE model. This thesis is comprised of five chapters where the main aim is to study the CTRE model on isolated time scales. Most of the models studied in economics are continuous or discrete. Discrete models are more preferable by economists since they give more meaningful and accurate results. Discrete models only contain uniform time domains. Time scale calculus enables us to study on m-periodic time domains as well as non periodic time domains. In the first chapter, we give basics of time scales calculus and stochastic calculus. The second chapter is the brief introduction to rational expectations and the CTRE model. Moreover, many other solution techniques are examined in this chapter. After we introduce the necessary background, in the third chapter we construct the CTRE Model on isolated time scales. Then we give the general solution of this model in terms of martingales. We continue our work with defining the linear system and higher order CTRE on isolated time scales. We use Putzer Algorithm to solve the system of the CTRE Model. Then, we examine the existence and uniqueness of the solution of the CTRE model. In the fourth chapter, we apply our solution algorithm developed in the previous chapter to models in Finance and stochastic growth models in Economics.