5. Independence
... Two events A and B are independent if ℙ( A∩B) = ℙ( A) ℙ( B) If both of the events have positive probability, then independence is equivalent to the statement that the conditional probability of one event given the other is the same as the unconditional probability of the event: ℙ( A|| B) = ℙ( A) if ...
... Two events A and B are independent if ℙ( A∩B) = ℙ( A) ℙ( B) If both of the events have positive probability, then independence is equivalent to the statement that the conditional probability of one event given the other is the same as the unconditional probability of the event: ℙ( A|| B) = ℙ( A) if ...
10/9/14
... We can nevertheless ask questions about how closely the sequence (ϕj )j∈N resembles a sequence of independent identically distributed random variables. Do they obey a law of large numbers, a central limit theorem, are there large deviation estimates, etc. The Poincaré Recurrence theorem: Let T : ( ...
... We can nevertheless ask questions about how closely the sequence (ϕj )j∈N resembles a sequence of independent identically distributed random variables. Do they obey a law of large numbers, a central limit theorem, are there large deviation estimates, etc. The Poincaré Recurrence theorem: Let T : ( ...
Presentation
... and therefore you should not invest money that you cannot afford to lose. Nothing in this presentation is a recommendation to buy or sell currencies and Interbank FX is not liable for any loss or damage, including without limitation, any loss of profit which may arise directly or indirectly from the ...
... and therefore you should not invest money that you cannot afford to lose. Nothing in this presentation is a recommendation to buy or sell currencies and Interbank FX is not liable for any loss or damage, including without limitation, any loss of profit which may arise directly or indirectly from the ...
Induction and Probability - ANU School of Philosophy
... The principal merits of the view are clear enough. It allows us to maintain, contra Hume and other skeptics about induction, a vigorous distinction between rational and irrational inductive methods and inferences, and it acquires at least some measure of plausibility from the dismal failure of more ...
... The principal merits of the view are clear enough. It allows us to maintain, contra Hume and other skeptics about induction, a vigorous distinction between rational and irrational inductive methods and inferences, and it acquires at least some measure of plausibility from the dismal failure of more ...
Chapter 4 Introduction to Probability
... Let A, B, and C denote the events the first, second and third patients, respectively, are allergic to penicillin. Hence a) P (A and B and C) = P(A) P(B) P(C) = (.20) (.20) (.20) = .008 b) Let us define the following events: G = all three patients are allergic & H = at least one patient is not allerg ...
... Let A, B, and C denote the events the first, second and third patients, respectively, are allergic to penicillin. Hence a) P (A and B and C) = P(A) P(B) P(C) = (.20) (.20) (.20) = .008 b) Let us define the following events: G = all three patients are allergic & H = at least one patient is not allerg ...
Probability, chance and the probability of chance
... chance is an abstract construct. It is a useful abstraction all the same, because in writing P(X = 1 | θ) = θ , you are saying that your stake on the uncertain event (X = 1) is θ , were you to know θ . But no one can possibly tell you what θ is, and this is what leads us to the next section. But bef ...
... chance is an abstract construct. It is a useful abstraction all the same, because in writing P(X = 1 | θ) = θ , you are saying that your stake on the uncertain event (X = 1) is θ , were you to know θ . But no one can possibly tell you what θ is, and this is what leads us to the next section. But bef ...
Estimating probabilities from counts with a prior of uncertain reliability
... πi = 1/k, which amounts to asserting that, as far as we know, all possible observations are equally likely. At other times, we may know some some more detailed approximation to the distribution θ. For example, we wish to estimate the probabilities of substituting a pair of amino acid residues by ano ...
... πi = 1/k, which amounts to asserting that, as far as we know, all possible observations are equally likely. At other times, we may know some some more detailed approximation to the distribution θ. For example, we wish to estimate the probabilities of substituting a pair of amino acid residues by ano ...
Ars Conjectandi
Ars Conjectandi (Latin for The Art of Conjecturing) is a book on combinatorics and mathematical probability written by Jakob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, apart from many combinatorial topics, many central ideas in probability theory, such as the very first version of the law of large numbers: indeed, it is widely regarded as the founding work of that subject. It also addressed problems that today are classified in the twelvefold way, and added to the subjects; consequently, it has been dubbed an important historical landmark in not only probability but all combinatorics by a plethora of mathematical historians. The importance of this early work had a large impact on both contemporary and later mathematicians; for example, Abraham de Moivre.Bernoulli wrote the text between 1684 and 1689, including the work of mathematicians such as Christiaan Huygens, Gerolamo Cardano, Pierre de Fermat, and Blaise Pascal. He incorporated fundamental combinatorial topics such as his theory of permutations and combinations—the aforementioned problems from the twelvefold way—as well as those more distantly connected to the burgeoning subject: the derivation and properties of the eponymous Bernoulli numbers, for instance. Core topics from probability, such as expected value, were also a significant portion of this important work.