infinite perimeter of the Koch snowflake and its finite - Dimes
... The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational methodology allowing one to execute numerical computations with infin ...
... The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational methodology allowing one to execute numerical computations with infin ...
QUANTITATIVE METHODS IN PSYCHOLOGY On the Probability of
... about probabilities. Why should there be such a fundamental tendency to confuse the posterior probability that the null hypothesis is true, given that it has been rejected, with the prior probability that the null hypothesis will be rejected, given that it is true? Kahneman and Tversky (1973) demons ...
... about probabilities. Why should there be such a fundamental tendency to confuse the posterior probability that the null hypothesis is true, given that it has been rejected, with the prior probability that the null hypothesis will be rejected, given that it is true? Kahneman and Tversky (1973) demons ...
BROWNIAN MOTION AND THE STRONG MARKOV PROPERTY
... εi = {∅, En , Ω\En , Ω}. At this point, we will define the concept of an expectation formally. Doing so, however, first requires an understanding of what taking an integral with respect to a measure means exactly. It will be beyond the scope of this paper to develop this theory in sufficient detail. ...
... εi = {∅, En , Ω\En , Ω}. At this point, we will define the concept of an expectation formally. Doing so, however, first requires an understanding of what taking an integral with respect to a measure means exactly. It will be beyond the scope of this paper to develop this theory in sufficient detail. ...
Math 6710 lecture notes
... This is a typical application. We want to show some property Q holds for all sets in some σ-field F . So we show that the collection of all sets with the property Q (whatever it may be) is a λ-system. Then we find a collection of sets P for which we know that Q holds, and show that it’s a π-system w ...
... This is a typical application. We want to show some property Q holds for all sets in some σ-field F . So we show that the collection of all sets with the property Q (whatever it may be) is a λ-system. Then we find a collection of sets P for which we know that Q holds, and show that it’s a π-system w ...
Chapter 8 Discrete probability and the laws of chance
... were then asked to record their results and to indicate how many heads they had obtained in this sequence of tosses. (Note that the order of the heads was not taken into account, only how many were obtained out of the 10 tosses.) The table shown below specifies the number, k, of heads (column 1), th ...
... were then asked to record their results and to indicate how many heads they had obtained in this sequence of tosses. (Note that the order of the heads was not taken into account, only how many were obtained out of the 10 tosses.) The table shown below specifies the number, k, of heads (column 1), th ...
Lower Bounds on Learning Random Structures with
... A deterministic finite acceptor M over the alphabet {0, 1} can be used to represent the set L(M ) ∩ {0, 1}n of accepted binary strings of length n. Gold [4] gave one of the earliest hardness results for learning DFAs, that finding a smallest DFA consistent with given positive and negative data is NP ...
... A deterministic finite acceptor M over the alphabet {0, 1} can be used to represent the set L(M ) ∩ {0, 1}n of accepted binary strings of length n. Gold [4] gave one of the earliest hardness results for learning DFAs, that finding a smallest DFA consistent with given positive and negative data is NP ...
1 Kroesus and the oracles
... In the middle of the 6th century B.C. the Persian realm started to expand. Kroisos (Kroesus), king of Lydia c:a 560–546 B.C., was worried by the developement. Lydia then covered what is now western Turkey, between the Hellespont and the river Halys which reaches almost from the coast opposite Cyprus ...
... In the middle of the 6th century B.C. the Persian realm started to expand. Kroisos (Kroesus), king of Lydia c:a 560–546 B.C., was worried by the developement. Lydia then covered what is now western Turkey, between the Hellespont and the river Halys which reaches almost from the coast opposite Cyprus ...
Almost sure lim sup behavior of bootstrapped means with
... proofs of those results. They also provided examples that show that the sizes of resampling required by their results to ensure almost sure (a.s.) convergence are not far from optimal. ...
... proofs of those results. They also provided examples that show that the sizes of resampling required by their results to ensure almost sure (a.s.) convergence are not far from optimal. ...
probability literacy, statistical literacy, adult numeracy, quantitative
... TOWARDS "PROBABILITY LITERACY" FOR ALL CITIZENS: BUILDING BLOCKS AND ...
... TOWARDS "PROBABILITY LITERACY" FOR ALL CITIZENS: BUILDING BLOCKS AND ...
2.10 Consider the following sequence of (0,1) random numbers
... (a) __________________ (b) _____________________ (c)_________________ (a) inverse transform, (b) convolution, (c) acceptance/rejection 2.3 Consider the multiplicative congruential generator with (a = 13, m = 64, and seeds X0 = 1,2,3,4) a) Does this generator achieve its maximum period for these para ...
... (a) __________________ (b) _____________________ (c)_________________ (a) inverse transform, (b) convolution, (c) acceptance/rejection 2.3 Consider the multiplicative congruential generator with (a = 13, m = 64, and seeds X0 = 1,2,3,4) a) Does this generator achieve its maximum period for these para ...
The "slippery" concept of probability: Reflections on possible
... Hawkins and Kapadia (1984) refer to these two approaches as the psychological and pedagogical approaches respectively, and note that there has been little synthesis between the two approaches. Furthermore, much of this research seems to be based on pupil observation rather than on actual empirical r ...
... Hawkins and Kapadia (1984) refer to these two approaches as the psychological and pedagogical approaches respectively, and note that there has been little synthesis between the two approaches. Furthermore, much of this research seems to be based on pupil observation rather than on actual empirical r ...
Comparing Infinite Sets - University of Arizona Math
... A finite set has a first element, second element, until it reaches its kth element. It did not keep going forever on the number of elements in the finite set. In an infinite set, you still have the first element, second element, and so on. However there is no last element because the infinite set wi ...
... A finite set has a first element, second element, until it reaches its kth element. It did not keep going forever on the number of elements in the finite set. In an infinite set, you still have the first element, second element, and so on. However there is no last element because the infinite set wi ...
Measuring fractals by infinite and infinitesimal numbers
... infinite radix described in this section evolves the idea of separate count of units with different exponents used in traditional positional systems to the case of infinite and infinitesimal numbers. The infinite radix of the new system is introduced as the number of elements of the set N of natural ...
... infinite radix described in this section evolves the idea of separate count of units with different exponents used in traditional positional systems to the case of infinite and infinitesimal numbers. The infinite radix of the new system is introduced as the number of elements of the set N of natural ...
Preprint.
... we discuss the earlier approaches in this field. We will also give a simple necessary and sufficient criterion for the existence of a strictly stationary solution to (1.1) and (1.5) with finite second moments. Our results are stated in Section 2. In Section 3 we give applications of the FCLT for sta ...
... we discuss the earlier approaches in this field. We will also give a simple necessary and sufficient criterion for the existence of a strictly stationary solution to (1.1) and (1.5) with finite second moments. Our results are stated in Section 2. In Section 3 we give applications of the FCLT for sta ...
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.