Reducing belief simpliciter to degrees of belief
... qualitative description to be a coarse-grained version of the quantitative description: which qualitative description applies should depend only on what quantitative description applies, maybe supplemented by ...
... qualitative description to be a coarse-grained version of the quantitative description: which qualitative description applies should depend only on what quantitative description applies, maybe supplemented by ...
Probability, Random Processes, and Ergodic Properties
... at least not before they need some of the material in their research. In addition, many of the existing mathematical texts on the subject are hard for this audience to follow, and the emphasis is not well matched to engineering applications. A notable exception is Ash’s excellent text [1], which was ...
... at least not before they need some of the material in their research. In addition, many of the existing mathematical texts on the subject are hard for this audience to follow, and the emphasis is not well matched to engineering applications. A notable exception is Ash’s excellent text [1], which was ...
Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute
... Let us look at the proof of this one. For convenience, let me take x to be a continuous random variable, let x be a continuous random variable with say probability density function f x. Let us write down expectation of modulus x to the power s, this is equal to integral minus infinity to infinity mo ...
... Let us look at the proof of this one. For convenience, let me take x to be a continuous random variable, let x be a continuous random variable with say probability density function f x. Let us write down expectation of modulus x to the power s, this is equal to integral minus infinity to infinity mo ...
Conditional Prediction without a Coarsening at Random condition
... There are many situations in which a statistician receives incomplete data and still has to reach conclusions about this data. One possibility of incomplete data is coarse data. This means that one does not observe the real outcome of a random event, but only a subset of all possible outcomes. An ex ...
... There are many situations in which a statistician receives incomplete data and still has to reach conclusions about this data. One possibility of incomplete data is coarse data. This means that one does not observe the real outcome of a random event, but only a subset of all possible outcomes. An ex ...
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 ...
Essentials of Stochastic Processes
... number of physical, biological, economic, and social phenomena that can be modeled in this way, and (ii) there is a well-developed theory that allows us to do computations. We begin with a famous example, then describe the property that is the defining feature of Markov chains Example 1.1. Gambler’s ...
... number of physical, biological, economic, and social phenomena that can be modeled in this way, and (ii) there is a well-developed theory that allows us to do computations. We begin with a famous example, then describe the property that is the defining feature of Markov chains Example 1.1. Gambler’s ...
WEAK AND STRONG LAWS OF LARGE NUMBERS FOR
... about something, say, the value that a random variable X assumes in a set of possible values X . Then a bounded real-valued function on X is called a gamble on X , and the set of all gambles on X is denoted by L (X ). We interpret a gamble as an uncertain reward: if the value of the random variable ...
... about something, say, the value that a random variable X assumes in a set of possible values X . Then a bounded real-valued function on X is called a gamble on X , and the set of all gambles on X is denoted by L (X ). We interpret a gamble as an uncertain reward: if the value of the random variable ...
Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.In this context, ""almost surely"" is a mathematical term with a precise meaning, and the ""monkey"" is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. One of the earliest instances of the use of the ""monkey metaphor"" is that of French mathematician Émile Borel in 1913, but the first instance may be even earlier. The relevance of the theorem is questionable—the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero). It should also be noted that real monkeys don't produce uniformly random output, which means that an actual monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text.Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters. In the early 20th century, Émile Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics.