Chapter 4 Dependent Random Variables
... Proof. All are easy consequences of properties of conditional expectations. Property (iii) perhaps needs an explanation. If E[|fn − f |] → 0 by the properties of conditional expectation E[|E{fn |Σ} − E{f |Σ}| → 0. Property (iii) is an easy consequence of this. The problem with the above theorem is t ...
... Proof. All are easy consequences of properties of conditional expectations. Property (iii) perhaps needs an explanation. If E[|fn − f |] → 0 by the properties of conditional expectation E[|E{fn |Σ} − E{f |Σ}| → 0. Property (iii) is an easy consequence of this. The problem with the above theorem is t ...
Probabilistic Limit Theorems
... with probability 1. Moreover, the set of limit points of the sequence (Sn =an )n1 is almost surely equal to the interval [,; +] (we then say that X satis es the LIL). With the exception of the last statement on the LIL these statements may be shown to easily extend to nite dimensional random var ...
... with probability 1. Moreover, the set of limit points of the sequence (Sn =an )n1 is almost surely equal to the interval [,; +] (we then say that X satis es the LIL). With the exception of the last statement on the LIL these statements may be shown to easily extend to nite dimensional random var ...
Dp2007-08 - Research portal
... and assume that i and k share a group and that j and k share a group. Then, the probability that i and j also have a common group depends on the number of groups that the common neighbor k belongs to. Indeed, the fewer groups k belongs to, the more likely it is that i and j in fact share the same gr ...
... and assume that i and k share a group and that j and k share a group. Then, the probability that i and j also have a common group depends on the number of groups that the common neighbor k belongs to. Indeed, the fewer groups k belongs to, the more likely it is that i and j in fact share the same gr ...
Module 5 - University of Pittsburgh
... The total number of nodes with degree k is npk Hence the probability that a neighbor of a node has degree k is: ...
... The total number of nodes with degree k is npk Hence the probability that a neighbor of a node has degree k is: ...
Powerpoint 3: Strings and arrays
... their five favorite numbers, and store those numbers in an array. • Modify your program to ask the user for a number n, and then ask the user for their n favorite numbers, and store those numbers in an array. ...
... their five favorite numbers, and store those numbers in an array. • Modify your program to ask the user for a number n, and then ask the user for their n favorite numbers, and store those numbers in an array. ...
Reasoning with Limited Resources and Assigning Probabilities to
... truths. But if mathematics is in the business of discovering truths, then it plays in the same field with other scientific inquiries. The fact that an investigation is likely to extend our computational/deductive reach, and that rationality demands that we accept the deductive consequences of our be ...
... truths. But if mathematics is in the business of discovering truths, then it plays in the same field with other scientific inquiries. The fact that an investigation is likely to extend our computational/deductive reach, and that rationality demands that we accept the deductive consequences of our be ...
Scalable Analysis and Design of Ad Hoc Networks Via Random
... and the probability distribution of ω(Gn,p ) sharply concentrates on two consequtive integers. For a finite graph this holds almost surely, which means that the fraction of all graphs that deviate from the formula tends to 0 as n grows. Thus, for a random graph, one can practically determine the desi ...
... and the probability distribution of ω(Gn,p ) sharply concentrates on two consequtive integers. For a finite graph this holds almost surely, which means that the fraction of all graphs that deviate from the formula tends to 0 as n grows. Thus, for a random graph, one can practically determine the desi ...
Reasoning with Limited Resources and
... truths. But if mathematics is in the business of discovering truths, then it plays in the same field with other scientific inquiries. The fact that an investigation is likely to extend our computational/deductive reach, and that rationality demands that we accept the deductive consequences of our be ...
... truths. But if mathematics is in the business of discovering truths, then it plays in the same field with other scientific inquiries. The fact that an investigation is likely to extend our computational/deductive reach, and that rationality demands that we accept the deductive consequences of our be ...
a critical evaluation of comparative probability - Philsci
... that of establishing a probability given certain evidences. 8. From the examples we proposed it seems that it is often possible to establish a comparison between probabilities, but not to determine their quantitative value. Indeed there are few cases where the probability can be evaluated quantitati ...
... that of establishing a probability given certain evidences. 8. From the examples we proposed it seems that it is often possible to establish a comparison between probabilities, but not to determine their quantitative value. Indeed there are few cases where the probability can be evaluated quantitati ...
AP 7.5B Notes
... Shape: The probability distribution of X is skewed to the right. It is more likely to have 0, 1, or 2 children with type O blood than a larger value. Center: The median number of children with type O blood is 1. Based on our formula for the mean (find the expected value): ...
... Shape: The probability distribution of X is skewed to the right. It is more likely to have 0, 1, or 2 children with type O blood than a larger value. Center: The median number of children with type O blood is 1. Based on our formula for the mean (find the expected value): ...
THE EVALUATION OF EXPERIMENTAL RESULTS
... prediction. At this point, a geneticist would begin the search for a reasonable explanation: the original observations are always open to scrutiny; selection may have acted against one of the phenotypes, so that some of those individuals died, thus leading to fewer representatives of this class than ...
... prediction. At this point, a geneticist would begin the search for a reasonable explanation: the original observations are always open to scrutiny; selection may have acted against one of the phenotypes, so that some of those individuals died, thus leading to fewer representatives of this class than ...
CHAPTER I - Mathematics - University of Michigan
... We can use Borel-Cantelli (b) to prove recurrence for the standard random walk on the integers Z. Thus let the Xj , j = 1, 2, .., be Bernoulli variables taking the values ±1 with equal probability 1/2. Then SN = X1 + · · · + XN is the position after N steps of the standard random walk on Z starting ...
... We can use Borel-Cantelli (b) to prove recurrence for the standard random walk on the integers Z. Thus let the Xj , j = 1, 2, .., be Bernoulli variables taking the values ±1 with equal probability 1/2. Then SN = X1 + · · · + XN is the position after N steps of the standard random walk on Z starting ...
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.