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Concept# Probability

Summary

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).
These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer scien

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Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These top

Probability theory

Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the conc

Statistics

Statistics (from German: Statistik, "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and present

The financial crisis of 2007-2009 drew attention to the essential role of banks for the macroeconomy and to the importance of having a resilient financial sector. A vulnerability in the financial sector spills over to the real economy and can drive it into a deep recession. Following the financial crisis, policy makers have issued rules and regulations aimed at strengthening the banks. My dissertation considers some of those measures, namely bank liquidity regulation and deposit insurance, and tries to assess their effectiveness in stabilizing the financial sector and the economy in general.
The first chapter (joint with Prof. Luisa Lambertini) develops a model with regulated banks and a hedge fund to analyze the behavior of wholesale funding and the macroeconomic consequences of liquidity regulation. Banks raise deposits and subordinated wholesale funding from the hedge fund. Wholesale funding amplifies shocks: it is curtailed in economic downturns to avoid leveraging up and risk-taking by banks, further depressing credit and economic activity. By making banks safer, liquidity regulation increases wholesale funding and bank loans at the steady state. However, flat liquidity regulation requiring banks to hold a constant fraction of liquid assets can have unintended consequences and increase macroeconomic volatility. Our model further shows that cyclically adjusted regulation stabilizes the response of the economy to shocks. We also provide empirical evidence suggesting that liquidity helps contain the contraction in wholesale funding and loans during times of funding stress.
The second chapter explores the macroeconomic and welfare implications of harmonizing or joining deposit insurance in the Eurozone. I develop a two-country model to represent the core and periphery countries in the Eurozone. The model features fragile banks that can be subject to endogenous and costly runs. The two countries are financially integrated through an international interbank market. I introduce deposit insurance and compare different insurance regimes in steady state and along the business cycle. The model emphasizes that financial integration implies co-movement in asset prices and the transmission of shocks across the border. I find that when deposit insurance has lower coverage in the periphery than in the core, financial stability is undermined in both regions due to higher asset price volatility and a higher probability of deposit runs. Hence, either harmonizing or joining deposit insurance leads to substantial welfare benefits for both countries.
In the third chapter, I develop a macroeconomic model with multiple equilibria and bank runs to analyze deposit insurance. A bank run is a sunspot equilibrium, and it is caused by a self-fulfilling panic of all depositors. The model features a unique no-run equilibrium when macroeconomic conditions are good, but an unexpected negative shock may open up the possibility of a run equilibrium. A bank run is a rare but costly event that drives the economy in a long and severe recession. I analyze the introduction and build-up of a deposit insurance fund and show that it reduces the cost and probability of bank runs. The dynamic simulation of the model reveals that deposit insurance diminishes macroeconomic and financial instability following a technological shock.

Modern data storage systems are extremely large and consist of several tens or hundreds of nodes. In such systems, node failures are daily events, and safeguarding data from them poses a serious design challenge. The focus of this thesis is on the data reliability analysis of storage systems and, in particular, on the effect of different design choices and parameters on the system reliability. Data redundancy, in the form of replication or advanced erasure codes, is used to protect data from node failures. By storing redundant data across several nodes, the surviving redundant data on surviving nodes can be used to rebuild the data lost by the failed nodes if node failures occur. As these rebuild processes take a finite amount of time to complete, there exists a nonzero probability of additional node failures during rebuild, which eventually may lead to a situation in which some of the data have lost so much redundancy that they become irrecoverably lost from the system. The average time taken by the system to suffer an irrecoverable data loss, also known as the mean time to data loss (MTTDL), is a measure of data reliability that is commonly used to compare different redundancy schemes and to study the effect of various design parameters. The theoretical analysis of MTTDL, however, is a challenging problem for non-exponential real-world failure and rebuild time distributions and for general data placement schemes. To address this issue, a methodology for reliability analysis is developed in this thesis that is based on the probability of direct path to data loss during rebuild. The reliability analysis is detailed in the sense that it accounts for the rebuild times involved, the amounts of partially rebuilt data when additional nodes fail during rebuild, and the fact that modern systems use an intelligent rebuild process that will first rebuild the data having the least amount of redundancy left. Through rigorous arguments and simulations it is established that the methodology developed is well-suited for the reliability analysis of real-world data storage systems. Applying this methodology to data storage systems with different types of redundancy, various data placement schemes, and rebuild constraints, the effect of the design parameters on the system reliability is studied. When sufficient network bandwidth is available for rebuild processes, it is shown that spreading the redundant data corresponding to the data on each node across a higher number of other nodes and using a distributed and intelligent rebuild process will improve the system MTTDL. In particular, declustered placement, which corresponds to spreading the redundant data corresponding to each node equally across all other nodes of the system, is found to potentially have significantly higher MTTDL values than other placement schemes, especially for large storage systems. This implies that more reliable data storage systems can be designed merely by changing the data placement without compromising on the storage efficiency or performance. The effect of a limited network rebuild bandwidth on the system reliability is also analyzed, and it is shown that, for certain redundancy schemes, spreading redundant data across more number of nodes can actually have a detrimental effect on reliability. It is also shown that the MTTDL values are invariant in a large class of node failure time distributions with the same mean. This class includes the exponential distribution as well as the real-world distributions, such as Weibull or gamma. This result implies that the system MTTDL will not be affected if the failure distribution is changed to a corresponding exponential one with the same mean. This observation is also of great importance because it suggests that the MTTDL results obtained in the literature by assuming exponential node failure distributions may still be valid for real-world storage systems despite the fact that real-world failure distributions are non-exponential. In contrast, it is shown that the MTTDL is sensitive to the node rebuild time distribution. A storage system reliability simulator is built to verify the theoretical results mentioned above. The simulator is sufficiently complex to perform all required failure events and rebuild tasks in a storage system, to use real-world failure and rebuild time distributions for scheduling failures and rebuilds, to take into account partial rebuilds when additional node failures occur, and to simulate different data placement schemes and compare their reliability. The simulation results are found to match the theoretical predictions with high confidence for a wide range of system parameters, thereby validating the methodology of reliability analysis developed.

National Research Council of the National Academies, Prospective Evaluation of Applied Energy Research and Development at DOE (Phase One): A First Look Forward (2005) proposes a cost-benefit methodology to evaluate U.S. Department of Energy’s Research, Development, and Demonstration (RD&D) programs. This paper develops the methodology for nuclear energy programs. The RD&D process is analyzed in two stages with two success probabilities: (1) that the technology will transition from the R&D Stage to the Prototype Demonstration Stage, and (2) that the technology will be adopted commercially. It models discounted expected total benefits of an RD&D program as a function of the levels of funding, stage durations, the probabilities of success, and spillovers to other technologies. Project duration is an exponential function of funding. Project success is a logistic function of funding and uncertainty. Spillovers are linear functions of funding at each stage. This specification allows calculation of the marginal effects of changes in funding on discounted expected total benefits for a single technology. The paper uses this method to offer an optimal allocation of pre-prototype R&D funding in the development of the Generation IV International Forum’s advanced nuclear energy systems under a specific parameterization and funding constraint.

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This course focuses on dynamic models of random phenomena, and in particular, the most popular classes of such models: Markov chains and Markov decision processes. We will also study applications in queuing theory, finance, project management, etc.

Le cours présente les notions de base de la théorie des probabilités et de l'inférence statistique. L'accent est mis sur les concepts principaux ainsi que les méthodes les plus utilisées.

Le cours présente les notions de base de la théorie des probabilités et de l'inférence statistique. L'accent est mis sur les concepts principaux ainsi que les méthodes les plus utilisées.

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