Probability

Complete with a modal expressing probability: 1) I’m sure he is at home. = He __________________________ be at home. 2) There is a slight chance that she is absent. = She _______________________be absent. 3) It’s impossible she is not cheating. = He______________________ be cheating. 4) He is very peculiar, may be he doesn’t like chocolate. = He _________________not like chocolate. 5) As I know her logically she is sleeping now. = She___________________________

|-0.224 |-0.198 |5.5246 |0.478 | |ARCH Test: | | |F-statistic |0.109283 | Probability |0.896845 | | |Obs*R-squared |0.239371 | Probability |0.887199 | | | | | | | | |Dependent Variable: RESID^2 |

. . . . . . . . . . . 1.1 Probability foundations . . . . . . . . Basic set theory Disjointness Partition Sigma algebra Probability function Probability space Counting Conditional probability Bayes’ Rule Independence of events Mutual independence of events Random variables . . . . . . . . . . . Random variable Random vector Measurability Smallest σ-field Independence of r.v.s Independence of random vectors Mean independence Cumulative distribution function Probability mass function Marginal pmf Conditional

consists of four risk neutral agents - an investor, a firm, a credit rating agency and the regulator (the government). The firm has a project with uncertain returns requiring fixed initiation costs. The firms have private information regarding the default probability of their projects. The firm does not have capital to initiate the project and hence, has to approach the investor. This investment is raised by offering debt. The investor is rational and has Bayesian beliefs. The rating agency charges a fee to the

a higher tuition leads to more global knowledge gained by each student. The parameter θ measures the learning ability of each student. Each student knows his/her own learning ability. Across the country, θ is described by a probability density function p(.) and a probability distribution function P(.) over the range [θ , θ ]. University D charges each student tuition and fees of d. In addition, the domestic government adds a subsidy s to each student, which is assumed to be exogenously given. (In

capital accumulation Comparing countries like Belgium and Italy shows that it is reasonnable to believe that countries with a large public debt can be in a situation of multiple equilibria, some of them with a small probability of default and low interest rates ; others with a large probability of default and high interest rates. We illustrate that idea, and analyze the possible effects of the risk of default on capital accumulation (hence on the incentive to default) ; those effects result from the expectation

1483–1522. 21. Matthews, P. (1988) J. Theor. Prob. 1, 411–423. 22. Fill, J. (1995) Theor. Comp. Sci., in press. 23. Diaconis, P., Fill, J. & Pitman, J. (1992) Comb. Prob. Comp. 1, 135–155. 24. Diaconis, P. (1986) Group Representations in Probability and Statistics (IMS, Hayward, CA). 25. Dobrow, R. & Fill, J. (1995) Ann. Appl. Prob. 5, 20–36. 26. Saloff-Coste, L. (1994) Math. Zeit. 217, 641–677. 27. Belsley, E. (1993) Ph.D. thesis (Harvard Univ., Cambridge,MA). 28. Silver, J. (1995)

imported goods from New Zealand and needs 100,000 New Zealand dollars 180 days from now.  It is trying to determine whether to hedge this position.  Loras has developed the following probability distribution for the New Zealand dollar:   Possible Value of New Zealand Dollar in 180 Days Probability $.40 5% .45 10% .48 30% .50 30% .53 20% .55 5% The 180-day forward rate of the New Zealand dollar

intuitive and appropriate risk measure (leading to the mean-variance trade-off). The same year, Roy (1952) claimed that “the optimal bundle of assets (investment) for investors who employ the safety first principle is the portfolio that minimizes the probability of disaster”. Roy A. D. (1952), Safety first and the holding of assets, Econometrica, 20, 431-449. Markowitz H. M. (1952), Portfolio selection, Journal of Finance, 7, 77-91. 23 ´ ´ Arthur CHARPENTIER - econometrie de la finance (2008/2009)

organizations, and developing interdisciplinary projects are strongly encouraged. The major components that the BSc offer follow what we have seen in our 2 Year University Diploma: * Mathematics * Linear Algebra * Statistics / Probability * Programming * Programming and Problem Solving * Artificial Intelligence * Programming Languages * Compiler Construction * Computer Network There are therefore numerous courses which are common with