Lecture 2
... Towards this, consider the Cartesian product space = 1 2 generated from 1 and 2 such that if 1 and 2 , then every in is an ordered pair of the form = (, ). To arrive at a probability model we need to define the combined trio (, F, P). ...
... Towards this, consider the Cartesian product space = 1 2 generated from 1 and 2 such that if 1 and 2 , then every in is an ordered pair of the form = (, ). To arrive at a probability model we need to define the combined trio (, F, P). ...
Subjectivistic Interpretations of Probability
... Degrees of belief are to be interpreted behavioristically. Ramsey first proposed that degrees of belief be measured by betting odds: if one is willing to bet at odds of 1:5 on the occurrence of a three on the roll of a die, but at no higher odds, then one's degree of belief is 1/(1 5) = +.As Ramsey ...
... Degrees of belief are to be interpreted behavioristically. Ramsey first proposed that degrees of belief be measured by betting odds: if one is willing to bet at odds of 1:5 on the occurrence of a three on the roll of a die, but at no higher odds, then one's degree of belief is 1/(1 5) = +.As Ramsey ...
// DOES RANDOM MEAN PURE CHANCE? It is hard to know if
... The news article cited above is an example of how we associate randomness to a tragic story about five people being shot to death, apparently at random, in an exclusive suburb of Washington, DC. All incidents were relatively close in time and location. The word random is used several times in the CN ...
... The news article cited above is an example of how we associate randomness to a tragic story about five people being shot to death, apparently at random, in an exclusive suburb of Washington, DC. All incidents were relatively close in time and location. The word random is used several times in the CN ...
2008 – 2009 Log1 Contest Round 3 Theta Individual Name: 4 points
... The remainder can be found by division (synthetic or otherwise) or by the remainder theorem which states the remainder when P(x) is divided by (x-a) is P(a). So, using x=1, we get a remainder of 14. Since 3397 = 2449 + 948, and number that divides 3397 and 2449 must divide 948, therefore the gcf of ...
... The remainder can be found by division (synthetic or otherwise) or by the remainder theorem which states the remainder when P(x) is divided by (x-a) is P(a). So, using x=1, we get a remainder of 14. Since 3397 = 2449 + 948, and number that divides 3397 and 2449 must divide 948, therefore the gcf of ...
Document
... Everybody moves to the room with number twice that of their current room. All the odd numbered rooms are now free and he uses them to accommodate the infinite number of people on the bus ...
... Everybody moves to the room with number twice that of their current room. All the odd numbered rooms are now free and he uses them to accommodate the infinite number of people on the bus ...
10.4: Probabilistic Reasoning: Rules of Probability
... • And strength in turn was characterized in the last lecture in terms of probability: A strong argument is one in which it is probable (but not necessary) that if the premises are true, then the conclusion is true. ...
... • And strength in turn was characterized in the last lecture in terms of probability: A strong argument is one in which it is probable (but not necessary) that if the premises are true, then the conclusion is true. ...
Random Variables
... recording various pieces of numerical data for each trial. For example, when a patient visits the doctor’s office, their height, weight, temperature and blood pressure are recorded. These observations vary from patient to patient, hence they are called variables. We tend to call these random variabl ...
... recording various pieces of numerical data for each trial. For example, when a patient visits the doctor’s office, their height, weight, temperature and blood pressure are recorded. These observations vary from patient to patient, hence they are called variables. We tend to call these random variabl ...
6 The Basic Rules of Probability
... irrational, because so many of us come up with the wrong probability orderings. But perhaps people are merely careless! Perhaps most of us do not attend closely to the exact wording of the question, "Which of statements (aHf) are more probable, that is have the highest probability." Instead we think ...
... irrational, because so many of us come up with the wrong probability orderings. But perhaps people are merely careless! Perhaps most of us do not attend closely to the exact wording of the question, "Which of statements (aHf) are more probable, that is have the highest probability." Instead we think ...
com.1 The Compactness Theorem
... 1. Γ ϕ iff there is a finite Γ0 ⊆ Γ such that Γ0 ϕ. 2. Γ is satisfiable if and only if it is finitely satisfiable. Proof. We prove (2). If Γ is satisfiable, then there is a structure M such that M |= ϕ for all ϕ ∈ Γ . Of course, this M also satisfies every finite subset of Γ , so Γ is finitely s ...
... 1. Γ ϕ iff there is a finite Γ0 ⊆ Γ such that Γ0 ϕ. 2. Γ is satisfiable if and only if it is finitely satisfiable. Proof. We prove (2). If Γ is satisfiable, then there is a structure M such that M |= ϕ for all ϕ ∈ Γ . Of course, this M also satisfies every finite subset of Γ , so Γ is finitely s ...
Full text
... a difference of two sums (Proposition 2.1). This result enables us to express Fibonacci numbers, Tribonacci numbers, etc., and their generalizations as sums of weighted binomial coefficients. In probability literature (Feller [2]), the probability generating functions of waiting times of this type a ...
... a difference of two sums (Proposition 2.1). This result enables us to express Fibonacci numbers, Tribonacci numbers, etc., and their generalizations as sums of weighted binomial coefficients. In probability literature (Feller [2]), the probability generating functions of waiting times of this type a ...
Inequalities
... two random variables U and V are linearly dependent if there exist finite numbers a and b, not both 0, such that aU + bV = 0 with probability one. The proof of the following theorem is left to Exercise 5. Theorem 3, continued. Suppose kXkp < ∞ and kY kq < ∞. Then equality holds in (9) if and only if ...
... two random variables U and V are linearly dependent if there exist finite numbers a and b, not both 0, such that aU + bV = 0 with probability one. The proof of the following theorem is left to Exercise 5. Theorem 3, continued. Suppose kXkp < ∞ and kY kq < ∞. Then equality holds in (9) if and only if ...
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.