Chapter 5 - Elementary Probability Theory Historical Background
... being the “father” of probability theory. In the twentieth century a coherent mathematical theory of probability was developed through people such as Chebyshev, Markov, and Kolmogorov. Probability The study of probability is concerned with random phenomena. Even though we cannot be certain whether a ...
... being the “father” of probability theory. In the twentieth century a coherent mathematical theory of probability was developed through people such as Chebyshev, Markov, and Kolmogorov. Probability The study of probability is concerned with random phenomena. Even though we cannot be certain whether a ...
Title A characterization of contiguous probability
... that lim m(l—<£φm)=σ2. If σ2—0, then the theorem immediately follows from Lemma 4.2. Thus, assume that σ 2 >0. Since (A.I) implies (2) in Lemma 4.1, it is enough to show that the conditions (3) and (4) in Lemma 4.1 are satisfied. From (A.I) and (C.3), we have ...
... that lim m(l—<£φm)=σ2. If σ2—0, then the theorem immediately follows from Lemma 4.2. Thus, assume that σ 2 >0. Since (A.I) implies (2) in Lemma 4.1, it is enough to show that the conditions (3) and (4) in Lemma 4.1 are satisfied. From (A.I) and (C.3), we have ...
UNCERTAINTY THEORIES: A UNIFIED VIEW
... 1. Randomness: capturing variability through repeated observations. 2. Partial knowledge: because of information is often lacking, knowledge about issues of interest is generally not perfect. These two situations are not mutually exclusive. ...
... 1. Randomness: capturing variability through repeated observations. 2. Partial knowledge: because of information is often lacking, knowledge about issues of interest is generally not perfect. These two situations are not mutually exclusive. ...
1. Outline (1) Basic Graph Theory and graph coloring (2) Pigeonhole
... bounds (e.g. giving polynomial lower bounds of any fixed degree), but nothing reaching cn for any c > 1. This is achieved only by the Erdös Probability Method and the following two facts: The probability of the union of events is at most the sum of their probabilities with equality iff the events a ...
... bounds (e.g. giving polynomial lower bounds of any fixed degree), but nothing reaching cn for any c > 1. This is achieved only by the Erdös Probability Method and the following two facts: The probability of the union of events is at most the sum of their probabilities with equality iff the events a ...
Chapter 10 Monte Carlo Methods
... 2b − 1. Selecting M = 2b enables the largest possible period to be obtained for the computer system being used. Alternatively, M can be selected as a large prime number compatible with the computer word size. Once M is selected the value of A must satisfy 0 < A < M . The sequence of values {Xi } are ...
... 2b − 1. Selecting M = 2b enables the largest possible period to be obtained for the computer system being used. Alternatively, M can be selected as a large prime number compatible with the computer word size. Once M is selected the value of A must satisfy 0 < A < M . The sequence of values {Xi } are ...
Notes from Week 9: Multi-Armed Bandit Problems II 1 Information
... we gain by observing Y alone, plus the additional amount of certainty we gain by observing X, conditional on Y . The KL-divergence of two distributions can be thought of as a measure of their statistical distinguishability. We will need three lemmas concerning KL-divergence. The first lemma asserts ...
... we gain by observing Y alone, plus the additional amount of certainty we gain by observing X, conditional on Y . The KL-divergence of two distributions can be thought of as a measure of their statistical distinguishability. We will need three lemmas concerning KL-divergence. The first lemma asserts ...
PowerPoint - Dr. Justin Bateh
... die 10 times and count how many times the number 6 shows up. The number of trials is n = 10, and the probability that a 6 will show up is p = 1/6 = 0.1667 Symbolically, we can say X~Binomial(n=10, p=1/6), which is read as "X follows a binomial distribution with n = 10, and p = 1/6" ...
... die 10 times and count how many times the number 6 shows up. The number of trials is n = 10, and the probability that a 6 will show up is p = 1/6 = 0.1667 Symbolically, we can say X~Binomial(n=10, p=1/6), which is read as "X follows a binomial distribution with n = 10, and p = 1/6" ...
CMP3_G7_MS_ACE1
... times or tossing three coins at once does have the same number of equally likely outcomes. The outcomes include HHH, TTT, THT, HTH, TTH, HHT, THH, and HTT. Note: Some students may answer no for this question, which is fine as long as their reasoning is correct. They may say that the outcomes for tos ...
... times or tossing three coins at once does have the same number of equally likely outcomes. The outcomes include HHH, TTT, THT, HTH, TTH, HHT, THH, and HTT. Note: Some students may answer no for this question, which is fine as long as their reasoning is correct. They may say that the outcomes for tos ...
Notes on Infinite Sets
... In this case, for f: N Æ {0,1}*, we can say for each n≥0 and 2n–1 ≤ m ≤ 2n+1–2, define f(m) as the n-bit binary expansion of m–(2n–1). This provides a function f: N Æ {0,1}* that we can show is 1-1 and onto. Since each integer m falls uniquely between two successive powers of 2, given m, there is on ...
... In this case, for f: N Æ {0,1}*, we can say for each n≥0 and 2n–1 ≤ m ≤ 2n+1–2, define f(m) as the n-bit binary expansion of m–(2n–1). This provides a function f: N Æ {0,1}* that we can show is 1-1 and onto. Since each integer m falls uniquely between two successive powers of 2, given m, there is on ...
Chapter 3 Finite and infinite sets
... You might think that the number of points in a square would be larger again than the number of points on the line. But this is not so: Theorem 3.1.15 The cardinality of the set of points in the unit square is the same as that of the set of points in the unit interval [0, 1] on the line. Proof This i ...
... You might think that the number of points in a square would be larger again than the number of points on the line. But this is not so: Theorem 3.1.15 The cardinality of the set of points in the unit square is the same as that of the set of points in the unit interval [0, 1] on the line. Proof This i ...
ACE HW
... 3. Bailey uses the results from an experiment to calculate the probability of each color of block being chosen from a bucket. He says P(red) = 35%, P(blue) = 45%, P(yellow) = 20%. Jarod uses theoretical probability because he knows how many of each color block is in the bucket. He says P(red) = 45%, ...
... 3. Bailey uses the results from an experiment to calculate the probability of each color of block being chosen from a bucket. He says P(red) = 35%, P(blue) = 45%, P(yellow) = 20%. Jarod uses theoretical probability because he knows how many of each color block is in the bucket. He says P(red) = 45%, ...
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.