scribe notes - people.csail.mit.edu
... First, fix two arbitrary elements x and y in Ω. We argue that px and py converge to the same distribution, which is the stationary distribution. Consider two copies (Xt )t and (Yt )t of the same Markov chain X , the first starting at X0 = x and the second at Y0 = y. Let (Xt , Yt ) be an “independent ...
... First, fix two arbitrary elements x and y in Ω. We argue that px and py converge to the same distribution, which is the stationary distribution. Consider two copies (Xt )t and (Yt )t of the same Markov chain X , the first starting at X0 = x and the second at Y0 = y. Let (Xt , Yt ) be an “independent ...
Some Early Analytic Number Theory
... That is, the right-hand side must equal the sum of the harmonic series. But we know this series diverges—it does not have a finite sum. Therefore our assumption that there is only a finite number N of primes must be incorrect, and the proof is complete. Now this proof of Euler’s was by no means the ...
... That is, the right-hand side must equal the sum of the harmonic series. But we know this series diverges—it does not have a finite sum. Therefore our assumption that there is only a finite number N of primes must be incorrect, and the proof is complete. Now this proof of Euler’s was by no means the ...
Set theory
... Note that we have separated the digits of x and y into groups by always going to the next nonzero digit, inclusive. Now we associate to (x, y) the number z (0,1] by writing down the first x-group, after that the first y-group, then the second x-group, and so on. Thus, in our example, we ...
... Note that we have separated the digits of x and y into groups by always going to the next nonzero digit, inclusive. Now we associate to (x, y) the number z (0,1] by writing down the first x-group, after that the first y-group, then the second x-group, and so on. Thus, in our example, we ...
Bayes for Beginners - Wellcome Trust Centre for Neuroimaging
... • P(Brown eyes|Male): (P(A|B) with A := Brown eyes, B := Male) 1. What is the probability that a person has brown eyes, ignoring everyone who is not a male? 2. Ratio: (being a male with brown eyes)/(being a male) 3. Probability ratio: probability that a person is both male and has brown eyes to the ...
... • P(Brown eyes|Male): (P(A|B) with A := Brown eyes, B := Male) 1. What is the probability that a person has brown eyes, ignoring everyone who is not a male? 2. Ratio: (being a male with brown eyes)/(being a male) 3. Probability ratio: probability that a person is both male and has brown eyes to the ...
A Continuous Analogue of the Upper Bound Theorem
... is, one which has a density function with respect to the Lebesgue measure; µ is called balanced about a point p if every hyperplane through p equipartitions µ (by absolute continuity, the hyperplane itself has µ-measure zero). For a nonnegative integer k, let s̃k (µ, 0) be the probability that the c ...
... is, one which has a density function with respect to the Lebesgue measure; µ is called balanced about a point p if every hyperplane through p equipartitions µ (by absolute continuity, the hyperplane itself has µ-measure zero). For a nonnegative integer k, let s̃k (µ, 0) be the probability that the c ...
![[math.NT] 4 Jul 2014 Counting carefree couples](http://s1.studyres.com/store/data/017506615_1-305ec41ac7cda1cd4e55d19bf25308d7-300x300.png)
[math.NT] 4 Jul 2014 Counting carefree couples
... We have ζ(2)2 K1 ≈ 1.15876 and ζ(2)3K2 ≈ 1.27627. Thus, there is a positive correlation between squarefreeness and coprimality. Let I3 (x) denote the number of triples (a, b, c) with a ≤ x, b ≤ x, c ≤ x such that (a, b) = (a, c) = (b, c) = 1. Schroeder [26, Section 4.4] claims that I3 (x) ∼ K2 x3 . ...
... We have ζ(2)2 K1 ≈ 1.15876 and ζ(2)3K2 ≈ 1.27627. Thus, there is a positive correlation between squarefreeness and coprimality. Let I3 (x) denote the number of triples (a, b, c) with a ≤ x, b ≤ x, c ≤ x such that (a, b) = (a, c) = (b, c) = 1. Schroeder [26, Section 4.4] claims that I3 (x) ∼ K2 x3 . ...
Probability/Data - Fall River Public Schools
... marbles.You choose a marble from the bag at random. 1. What is the probability the marble is yellow? The probability it is blue? The probability it is red? 2. What is the sum of the probabilities from part (1)? 3. What color is the marble most likely to be? 4. What is the probability the marble is n ...
... marbles.You choose a marble from the bag at random. 1. What is the probability the marble is yellow? The probability it is blue? The probability it is red? 2. What is the sum of the probabilities from part (1)? 3. What color is the marble most likely to be? 4. What is the probability the marble is n ...
cheneyslides
... related to a realistic programming system. For appropriate domains, we should be able to prove all the results of classical probability theory. The constructors and operations from the programming model allow us to relate probability distributions and random values. And there are decision procedures ...
... related to a realistic programming system. For appropriate domains, we should be able to prove all the results of classical probability theory. The constructors and operations from the programming model allow us to relate probability distributions and random values. And there are decision procedures ...
Full tex
... Corollary 3.1: Let (P, Q) be a relatively prime pair of integers with P > |Q + 1| and Q 6= 1 and {Um }, {Vm } the associated generalised Fibonacci and Lucas sequences. If, for a fixed λ > 2, the Q∞function 1f : N → N has the property that f (n + 1) ≥ (λ + 2)f (n), then the infinite product n=1 (1 + ...
... Corollary 3.1: Let (P, Q) be a relatively prime pair of integers with P > |Q + 1| and Q 6= 1 and {Um }, {Vm } the associated generalised Fibonacci and Lucas sequences. If, for a fixed λ > 2, the Q∞function 1f : N → N has the property that f (n + 1) ≥ (λ + 2)f (n), then the infinite product n=1 (1 + ...
MTH/STA 561 EXPONENTIAL PROBABILITY DISTRIBUTION As
... that is, F (y) = e y/ for any positive rational number y. By the right continuity of F (y), it follows that F (y) = e y/ for any positive real number y. Therefore, Y has an exponential distribution with parameter > 0. The reason that the geometric and exponential distribution should both have the me ...
... that is, F (y) = e y/ for any positive rational number y. By the right continuity of F (y), it follows that F (y) = e y/ for any positive real number y. Therefore, Y has an exponential distribution with parameter > 0. The reason that the geometric and exponential distribution should both have the me ...
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.