Big Outliers Versus Heavy Tails: what to use?
... From Polya Theorem we obtain Maxwell distribution for velocities of gas molecules basing on two natural properties of the space as isotropy and homogeneity only. Are there any models leading in a natural way to heavy-tailed distributions? Let us show, that strictly stable distributions may be also d ...
... From Polya Theorem we obtain Maxwell distribution for velocities of gas molecules basing on two natural properties of the space as isotropy and homogeneity only. Are there any models leading in a natural way to heavy-tailed distributions? Let us show, that strictly stable distributions may be also d ...
Applications of Mathematics 12
... 1. Recall from chapter one: a Binomial experiment satisfies the following four conditions: 1. The experiment consists of n identical trials. 2. Each trial results in one of the two outcomes, called success and failure. 3. The probability of success, denoted p , remains the same from trial to trial. ...
... 1. Recall from chapter one: a Binomial experiment satisfies the following four conditions: 1. The experiment consists of n identical trials. 2. Each trial results in one of the two outcomes, called success and failure. 3. The probability of success, denoted p , remains the same from trial to trial. ...
Compound Probability March 10, 2014
... We can consider drawing two socks from the drawer to be two separate events. When we pull out the first sock, the probability of it begin red is 6/17. Then, when we pull out the second sock, there are only 16 socks left in the drawer, 5 of which are red. Therefore, for the second sock, the probabili ...
... We can consider drawing two socks from the drawer to be two separate events. When we pull out the first sock, the probability of it begin red is 6/17. Then, when we pull out the second sock, there are only 16 socks left in the drawer, 5 of which are red. Therefore, for the second sock, the probabili ...
Chapter 3: Random Variables
... One way to get a random variable is to think about the reward for a bet. We agree to play the following game. I flip a coin. The coin has P (H) = p, P (T ) = 1 − p. If the coin comes up heads, you pay me $q; if the coin comes up tails, I pay you $r. The number of dollars that change hands is a rando ...
... One way to get a random variable is to think about the reward for a bet. We agree to play the following game. I flip a coin. The coin has P (H) = p, P (T ) = 1 − p. If the coin comes up heads, you pay me $q; if the coin comes up tails, I pay you $r. The number of dollars that change hands is a rando ...
Set Theory Digression
... and D when duplication is allowed and order is important. The result according to the formula is: n = 4, and x = 2, consequently the possible number of combinations is M24 = 42 = 16. To …nd the result we can also use a tree diagram. 2. Duplication is not permissible and Order is important (Permutati ...
... and D when duplication is allowed and order is important. The result according to the formula is: n = 4, and x = 2, consequently the possible number of combinations is M24 = 42 = 16. To …nd the result we can also use a tree diagram. 2. Duplication is not permissible and Order is important (Permutati ...
−1({2})) = P({hht, hth, thh}) =
... Example 3.1.3. A board game has a wheel that is to be spun periodically. The wheel can stop in one of ten equally likely spots. Four of these spots are red, three are blue, two are green, and one is black. Let X denote the color of the spot. Determine the distribution of X. The function X is defined ...
... Example 3.1.3. A board game has a wheel that is to be spun periodically. The wheel can stop in one of ten equally likely spots. Four of these spots are red, three are blue, two are green, and one is black. Let X denote the color of the spot. Determine the distribution of X. The function X is defined ...
Inference V: MCMC Methods - CS
... Problem: It is difficult to sample from P(X1, …. Xn |e ) We had to use likelihood weighting to reweigh our samples This introduced bias in estimation In some case, such as when the evidence is on leaves, these methods are inefficient ...
... Problem: It is difficult to sample from P(X1, …. Xn |e ) We had to use likelihood weighting to reweigh our samples This introduced bias in estimation In some case, such as when the evidence is on leaves, these methods are inefficient ...
Randomness
Randomness is the lack of pattern or predictability in events. A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination. Individual random events are by definition unpredictable, but in many cases the frequency of different outcomes over a large number of events (or ""trials"") is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as often as 4. In this view, randomness is a measure of uncertainty of an outcome, rather than haphazardness, and applies to concepts of chance, probability, and information entropy.The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in random sequences. A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness.Randomness is most often used in statistics to signify well-defined statistical properties. Monte Carlo methods, which rely on random input (such as from random number generators or pseudorandom number generators), are important techniques in science, as, for instance, in computational science. By analogy, quasi-Monte Carlo methods use quasirandom number generators.Random selection is a method of selecting items (often called units) from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, a random selection mechanism would choose a red marble with probability 1/10. Note that a random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen. That is, if the selection process is such that each member of a population, of say research subjects, has the same probability of being chosen then we can say the selection process is random.