Probabilistic thinking and probability literacy in the context of risk
... 1 the ability to balance between psychological and formal elements (a, e)); 2 the understanding that direct criteria for success are missing (b); 3 the ability to separate between randomness and causality (c); and 4 the ability to separate reflecting on a problem and making a decision (d). In the f ...
... 1 the ability to balance between psychological and formal elements (a, e)); 2 the understanding that direct criteria for success are missing (b); 3 the ability to separate between randomness and causality (c); and 4 the ability to separate reflecting on a problem and making a decision (d). In the f ...
problems_and_solutio..
... (c) what is the probability that there will be at least one mutual fund that outperforms the market for each of 15 consecutive calendar years during some 15-year interval during the next 50 years? To answer this question, let p(s, n) be the probability that one mutual fund will achieve a streak of s ...
... (c) what is the probability that there will be at least one mutual fund that outperforms the market for each of 15 consecutive calendar years during some 15-year interval during the next 50 years? To answer this question, let p(s, n) be the probability that one mutual fund will achieve a streak of s ...
Expectations, Variances, Covariances, and Sample Means
... Important Property : Cov(X; Y) = 0 ) Var(X + Y) = Var(X) + Var(Y): To see why this is the case, for simplicity let E(X) = E(Y ) = 0: Then for cov(X; Y ) = 0 : ...
... Important Property : Cov(X; Y) = 0 ) Var(X + Y) = Var(X) + Var(Y): To see why this is the case, for simplicity let E(X) = E(Y ) = 0: Then for cov(X; Y ) = 0 : ...
Large sample properties of Gibbs
... Theorem 1. Let p̃ be a Gibbs-type prior with prior guess P∗ = E[ p̃], whose support coincides with X, and assume condition (H) holds true. If (Xi )i≥1 is a sequence of independent and identically distributed random elements from P0 then the posterior converges weakly, a.s.-P0∞ , to a point mass at α ...
... Theorem 1. Let p̃ be a Gibbs-type prior with prior guess P∗ = E[ p̃], whose support coincides with X, and assume condition (H) holds true. If (Xi )i≥1 is a sequence of independent and identically distributed random elements from P0 then the posterior converges weakly, a.s.-P0∞ , to a point mass at α ...
Introduction to Probability and Statistics Eleventh Edition
... I. What’s in last lecture? Descriptive Statistics – Numerical Measures. Chapter 2. II. What's in this and the next two lectures? Experiment, Event, Sample space, Probability, Counting rules, Conditional probability, Bayes’s rule, random variables, mean, variance. Read Chapter 3. 5E Note 4 ...
... I. What’s in last lecture? Descriptive Statistics – Numerical Measures. Chapter 2. II. What's in this and the next two lectures? Experiment, Event, Sample space, Probability, Counting rules, Conditional probability, Bayes’s rule, random variables, mean, variance. Read Chapter 3. 5E Note 4 ...
Section 6-3
... Most important skill for using binomial distributions is the ability to recognize situations to which they do and don’t apply ...
... Most important skill for using binomial distributions is the ability to recognize situations to which they do and don’t apply ...
Finding the Probability Distribution for Failures
... two parameters: the mean µ and the standard deviation σ. The normal probability distribution is symmetric around its mean. In our impeller example, the mean is 10 years. The mean defines how long the impellers last on average, in other words the mean time between failures. The standard deviation fol ...
... two parameters: the mean µ and the standard deviation σ. The normal probability distribution is symmetric around its mean. In our impeller example, the mean is 10 years. The mean defines how long the impellers last on average, in other words the mean time between failures. The standard deviation fol ...
Combinatorial Description and Free Convolution
... Thus we will present here another approach to freeness, more combinatorial in nature, which puts the main emphasis on so called “free cumulants”. These are some polynomials in the moments which behave much nicer with respect to freeness than the moments. The nomenclature comes from classical probabi ...
... Thus we will present here another approach to freeness, more combinatorial in nature, which puts the main emphasis on so called “free cumulants”. These are some polynomials in the moments which behave much nicer with respect to freeness than the moments. The nomenclature comes from classical probabi ...
Game theory, duality, economic growth
... In Vienna, I had had the good fortune to be able to attend Karl Menger’s seminar. In its organization, this seminar was a marvel. The program was very loose, and people talked about every which thing. The result was that every week gave birth to a new idea, small or large, but always attractive. Bac ...
... In Vienna, I had had the good fortune to be able to attend Karl Menger’s seminar. In its organization, this seminar was a marvel. The program was very loose, and people talked about every which thing. The result was that every week gave birth to a new idea, small or large, but always attractive. Bac ...
Legal Decisions and the Reference-Class Problem
... to ignore half of what is important. And crucially, for present purposes, the sentencing in Shonubi and other legal cases is not a purely statistical problem; it is a decision-theory problem. This important aspect of decision making, legal or otherwise, has been overlooked by some commentators. Our ...
... to ignore half of what is important. And crucially, for present purposes, the sentencing in Shonubi and other legal cases is not a purely statistical problem; it is a decision-theory problem. This important aspect of decision making, legal or otherwise, has been overlooked by some commentators. Our ...
Combinatorial Probability
... The last equality is easy to prove: The number of ways of picking m objects out of n to take is the same as the number of ways of choosing n − m to leave behind. Of course, one can also check this directly from the formula in (2.3). Pascal’s triangle. The number of outcomes for coin tossing problems ...
... The last equality is easy to prove: The number of ways of picking m objects out of n to take is the same as the number of ways of choosing n − m to leave behind. Of course, one can also check this directly from the formula in (2.3). Pascal’s triangle. The number of outcomes for coin tossing problems ...
notes
... We can answer our quality and certainty questions by looking at the statistical distribution of χ2 : quality of fit is related to the distribution of the residual χ2 , and confidence in the parameters is related to the distribution of fitted parameters. For quality of fit: suppose we have a good mod ...
... We can answer our quality and certainty questions by looking at the statistical distribution of χ2 : quality of fit is related to the distribution of the residual χ2 , and confidence in the parameters is related to the distribution of fitted parameters. For quality of fit: suppose we have a good mod ...
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.