Old notes from a probability course taught by Professor Lawler
... Throughout this section we will assume that there is a probability space (Ω, A, P). All σ-algebras will be sub-σ-algebras of A, and all random variables will be defined on this space. Definition • Two events A, B are called independent if P(A ∩ B) = P(A)P(B). • A collection of events {Aα } is called ...
... Throughout this section we will assume that there is a probability space (Ω, A, P). All σ-algebras will be sub-σ-algebras of A, and all random variables will be defined on this space. Definition • Two events A, B are called independent if P(A ∩ B) = P(A)P(B). • A collection of events {Aα } is called ...
How to determine if a random graph with a fixed degree sequence
... results cannot be directly applied to real-world networks where hubs are present and for each particular network adhoc approaches are needed (see e.g. the Aiello-Chung-Lu model for the case of scale-free networks [1]). Another problem is that all the previous results apply to a sequence of degree se ...
... results cannot be directly applied to real-world networks where hubs are present and for each particular network adhoc approaches are needed (see e.g. the Aiello-Chung-Lu model for the case of scale-free networks [1]). Another problem is that all the previous results apply to a sequence of degree se ...
"Approximation Theory of Output Statistics,"
... channels we consider is very general. To keep the development as simple as possible we restrict attention to channels with finite input/output alphabets. However, most of the proofs do not rely on that assumption, and it is specifically pointed out when this is not the case. In addition to allowing ...
... channels we consider is very general. To keep the development as simple as possible we restrict attention to channels with finite input/output alphabets. However, most of the proofs do not rely on that assumption, and it is specifically pointed out when this is not the case. In addition to allowing ...
More Efficient PAC-learning of DNF with Membership Queries Under
... Jackson [15] gave the first polynomial-time PAC learning algorithm for DNF with membership queries under the uniform distribution. However, the algorithm’s time and sample complexity make it impractical for all but relatively small problems. The algorithm is also not particularly efficient in its us ...
... Jackson [15] gave the first polynomial-time PAC learning algorithm for DNF with membership queries under the uniform distribution. However, the algorithm’s time and sample complexity make it impractical for all but relatively small problems. The algorithm is also not particularly efficient in its us ...
Toward Evidence-Based Medical Statistics. 2: The
... conclude that the observed differences are not likely to be true (4, 20, 21). They conclude this not always because contradictory prior evidence outweighed the trial evidence but because the trial evidence, when measured properly, was not very strong in the first place. It also provides justificatio ...
... conclude that the observed differences are not likely to be true (4, 20, 21). They conclude this not always because contradictory prior evidence outweighed the trial evidence but because the trial evidence, when measured properly, was not very strong in the first place. It also provides justificatio ...
Impact of the Three Gorges Dam, the South–North Water Transfer
... chlorinity? Second, if the Three Gorges Dam, SNWTP and local abstractions planned for 2030–2035 had operated from 1950–1951 to 2013–2014, what would have been the probability of an intrusion of given duration and chlorinity? The difference in probabilities is ascribed to the three modifications to t ...
... chlorinity? Second, if the Three Gorges Dam, SNWTP and local abstractions planned for 2030–2035 had operated from 1950–1951 to 2013–2014, what would have been the probability of an intrusion of given duration and chlorinity? The difference in probabilities is ascribed to the three modifications to t ...
The probability that the hyperbolic random graph is connected
... properties of large but finite random geometric graphs. Here N points are sampled within a certain region of Rd following a certain distribution and any two of them are joined when their Euclidean distance is smaller than some threshold which, in general, is a function of N . In the last two decades ...
... properties of large but finite random geometric graphs. Here N points are sampled within a certain region of Rd following a certain distribution and any two of them are joined when their Euclidean distance is smaller than some threshold which, in general, is a function of N . In the last two decades ...
Possibility Theory and its Applications: Where Do we Stand
... and necessity measures) instead of only one. Besides, it is not additive and makes sense on ordinal structures. The name “Theory of Possibility” was coined by Zadeh [142], who was inspired by a paper by Gaines and Kohout [91]. In Zadeh’s view, possibility distributions were meant to provide a graded ...
... and necessity measures) instead of only one. Besides, it is not additive and makes sense on ordinal structures. The name “Theory of Possibility” was coined by Zadeh [142], who was inspired by a paper by Gaines and Kohout [91]. In Zadeh’s view, possibility distributions were meant to provide a graded ...
The entropy per coordinate of a random vector is highly constrained
... vector of any dimension with given density at the mode is shown to have a range of just 1. Uniform distributions on convex bodies are at the lower end of this range, the distribution with i.i.d. exponentially distributed coordinates is at the upper end, and the normal is exactly in the middle. Thus, ...
... vector of any dimension with given density at the mode is shown to have a range of just 1. Uniform distributions on convex bodies are at the lower end of this range, the distribution with i.i.d. exponentially distributed coordinates is at the upper end, and the normal is exactly in the middle. Thus, ...
On Worst-Case to Average-Case Reductions for NP Problems
... and general approach to prove a statement of the form: “If NP 6⊆ BPP then distributional NP contains intractable problems.” Cryptography versus NP-hardness. The seminal work of Diffie and Hellman [DH76] introducing public key cryptography asked if there exists a public key encryption scheme whose ha ...
... and general approach to prove a statement of the form: “If NP 6⊆ BPP then distributional NP contains intractable problems.” Cryptography versus NP-hardness. The seminal work of Diffie and Hellman [DH76] introducing public key cryptography asked if there exists a public key encryption scheme whose ha ...
Ars Conjectandi
Ars Conjectandi (Latin for The Art of Conjecturing) is a book on combinatorics and mathematical probability written by Jakob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, apart from many combinatorial topics, many central ideas in probability theory, such as the very first version of the law of large numbers: indeed, it is widely regarded as the founding work of that subject. It also addressed problems that today are classified in the twelvefold way, and added to the subjects; consequently, it has been dubbed an important historical landmark in not only probability but all combinatorics by a plethora of mathematical historians. The importance of this early work had a large impact on both contemporary and later mathematicians; for example, Abraham de Moivre.Bernoulli wrote the text between 1684 and 1689, including the work of mathematicians such as Christiaan Huygens, Gerolamo Cardano, Pierre de Fermat, and Blaise Pascal. He incorporated fundamental combinatorial topics such as his theory of permutations and combinations—the aforementioned problems from the twelvefold way—as well as those more distantly connected to the burgeoning subject: the derivation and properties of the eponymous Bernoulli numbers, for instance. Core topics from probability, such as expected value, were also a significant portion of this important work.