Wolfram Research Mathworld, Repository hosted at UIUC
(13.662 recursos)
Wolfram Research's MathWorld is a comprehensive and interactive mathematics encyclopedia intended for students, educators, math enthusiasts, and researchers. It has been assembled by internet encyclopedist Eric W.ÊWeisstein with assistance from the mathematics and internet communities. Like the discipline of mathematics, this site is continuously updated to include new material and incorporate new discoveries. This is a free service for the mathematical community provided by Wolfram Research, makers of Mathematica, with additional support from the National Science Foundation.
Mostrando recursos 1 - 20 de 57
1.
Bayes' Theorem -- from MathWorld - Weisstein, Eric W.
Let A and B_j be sets. Conditional probability requires that P(A\cap B_j) = P(A)P(B_j\vert A), where \cap denotes intersection ("and"), and also that P(A\cap B_j) = P(B_j\cap A) = P(B_j)P(A\vert B_j). Therefore, P(B_j\vert A) = {P(B_j)P(A\vert B_j) \over P(A)}. Now, let S \equiv \bigcup_{i=1}^N A_i, so A_i is an event in S and A_i\cap A_j=\varnothing for i\not= j, then A = A\cap S = A\cap \left({\,\bigcup_{i=1}^N A_i}\right)= \bigcup_{i=1}^N (A\cap A_i) P(A) =...
2.
Bonferroni's Inequalities -- from MathWorld - Weisstein, Eric W.
Let P(E_i) be the probability that E_i is true, and P\left({\bigcup_{i=1}^n E_i}\right) be the probability that at least one of E_1, E_2, ..., E_n is true. Then P\left({\,\bigcup_{i=1}^n E_i}\right)\leq \sum_{i=1}^n P(E_i). A slightly wider class of inequalities are also known as "Bonferroni inequalities." See also: Inclusion-Exclusion Principle
3.
Boole's Inequality -- from MathWorld - Weisstein, Eric W.
Let P(E_i) be the probability of an event E_i occurring. Then P\left({\,\bigcup_{i=1}^N E_i}\right)\leq \sum_{i=1}^N P(E_i), where \cup denotes the union. If E_i and E_j are disjoint sets for all i and j, then the inequality becomes an equality. A beautiful theorem that expresses the exact relationship between the probability of unions and probabilities of individual events is known as the inclusion-exclusion principle. See also: Disjoint Sets, Inclusion-Exclusion Principle, Union
4.
Borel-Cantelli Lemma -- from MathWorld - Weisstein, Eric W.
Let \{A_n\}_{n=0}^\infty be a sequence of events occurring with a certain probability distribution, and let A be the event consisting of the occurrence of a finite number of events A_n for n = 1, 2, .... Then the probability of an infinite number of the A_n occurring is zero if \sum_{n=1}^\infty P(A_n) < \infty. Equivalently, in the extreme case of P(A_n)=0 for all n, the probability that none of them occurs is 1 and, in particular, the probability of A that a finite number occur is also...
5.
Chuck-a-Luck -- from MathWorld - Weisstein, Eric W.
A gambling game played at carnivals in which a player may bet on any one of the numbers 1 through 6. Three dice are then rolled and, if his number appears, he receives back a multiple of his original stake equal to the number of dice on which it appears plus the original stake. Otherwise, he loses his stake. The expected loss per roll is then given by {{120\over 216}}\times 0+{{90\over 216}}... ...ver 6}} = {{17\over 216}} \approx 0.079.
6.
Coin Tossing -- from MathWorld - Weisstein, Eric W.
An idealized coin consists of a circular disk of zero thickness which, when thrown in the air and allowed to fall, will rest with either side face up ("heads" H or "tails" T) with equal probability. A coin is therefore a two-sided die. Despite slight differences between the sides and nonzero thickness of actual coins, the distribution of their tosses makes a good approximation to a p=1/2 Bernoulli distribution. There are, however, some rather counterintuitive properties of...
7.
Coincidence -- from MathWorld - Weisstein, Eric W.
A coincidence is a surprising concurrence of events, perceived as meaningfully related, with no apparent causal connection (Diaconis and Mosteller 1989). Given a large number of events, extremely unlikely coincidences are possible--and perhaps even common. To quote Sherlock Holmes, "Amid the action and reaction of so dense a swarm of humanity, every possible combination of events may be expected to take place, and many a little problem will be presented which may be striking and...
8.
Conditional Probability -- from MathWorld - Weisstein, Eric W.
The conditional probability of an event A assuming that B has occurred, denoted P(A\vert B), equals P(A\vert B) = {P(A\cap B)\over P(B)}, which can be proven directly using a Venn diagram. Multiplying through, this becomes P(A\vert B)P(B) = P(A\cap B), which can be generalized to P(A\cap B\cap C) =P(A)P(B\vert A)P(C\vert A\cap B). Rearranging (1) gives P(B\vert A) = {P(B\cap A)\over P(A)}. Solving (4) for P(B\cap A) = P(A\cap B) and plugging in to (1) gives P(A\vert B) = {P(A)P(B\vert...
9.
Coupon Collector's Problem -- from MathWorld - Weisstein, Eric W.
Let n objects be picked repeatedly with probability p_i that object i is picked on a given try, with \sum_i p_i=1. Find the earliest time at which all n objects have been picked at least once. See also: Birthday Problem
13.
Fermat's Principle of Conjunctive Probability -- from MathWorld - Weisstein, Eric W.
The probability that two events will both happen is hk, where h is the probability that the first event will happen, and k is the probability that the second event will happen when the first event is known to have happened. See also: Conditional Probability
15.
Gambler's Ruin -- from MathWorld - Weisstein, Eric W.
Let two players each have a finite number of pennies (say, n_1 for player one and n_2 for player two). Now, flip one of the pennies (from either player), with each player having 50% probability of winning, and give the penny to the winner. Now repeat the process until one player has all the pennies. If the process is repeated indefinitely, the probability that one of the two player will eventually lose all his pennies must be 100%. In fact, the chances P_1 and P_2 that players one and two,...
16.
Hazard Function -- from MathWorld - Weisstein, Eric W.
The hazard function (also known as the failure rate, hazard rate, or force of mortality) h(x) is the ratio of the probability function P(x) to the survival function S(x), given by h(x) = {P(x)\over S(x)} = {P(x)\over 1-D(x)}, where D(x) is the distribution function (Evans et al. 2000, p. 13). See also: Mills Ratio, Probability Function, Survival Function
18.
Independent Statistics -- from MathWorld - Weisstein, Eric W.
Two variates A and B are statistically independent iff the conditional probability P(A\vert B) of A given B satisfies P(A\vert B) = P(A), in which case the probability of A and B is just P(AB)=P(A\cap B) = P(A)P(B). Similarly, n events A_1, A_2, ..., A_n are independent iff P\left({\,\bigcap_{i=1}^n A_i}\right)= \prod_{i=1}^n P(A_i). Statistically independent variables are always uncorrelated, but the converse is not necessarily true. See also: Bayes' Formula, Conditional Probability,...
19.
Kolmogorov's Axioms -- from MathWorld - Weisstein, Eric W.
Let Q_i denote anything subject to weighting by a normalized linear scheme of weights that sum to unity in a set W. The Kolmogorov axioms state that 1. For every Q_i in W, there is a real number Q (the Kolmogorov weight of Q_i) such that 0 < Q(Q_i) < 1. 2. Q(Q_i) + Q(\bar Q_i) = 1, where \bar Q_i denotes the complement of Q_i in W. 3. For the mutually exclusive subsets Q_1, Q_2, ... in W, Q(Q_1\cup Q_2\cup Q_3\cup \dots) = Q(Q_1) + Q(Q_2) + Q(Q_3) + \dots. See also: Probability Axioms...
20.
Likelihood -- from MathWorld - Weisstein, Eric W.
The hypothetical probability that an event which has already occurred would yield a specific outcome. The concept differs from that of a probability in that a probability refers to the occurrence of future events, while a likelihood refers to past events with known outcomes. See also: Likelihood Function, Likelihood Ratio, Maximum Likelihood,Maximum Likelihood Estimator, Negative Likelihood Ratio, Probability
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