Why should one expect to find long runs of (non)
... A BSTRACT. Sondow et al have studied Ramanujan primes (RPs) and observed numerically that, while half of all primes are RPs asymptotically, one obtains runs of consecutives RPs (resp. non-RPs) which are statistically significantly longer than one would expect if one was tossing an unbiased coin. In ...
... A BSTRACT. Sondow et al have studied Ramanujan primes (RPs) and observed numerically that, while half of all primes are RPs asymptotically, one obtains runs of consecutives RPs (resp. non-RPs) which are statistically significantly longer than one would expect if one was tossing an unbiased coin. In ...
Set 10: Randomized Algorithms
... assumption that inputs exhibit some distribution, make your own input distribution by, say, permuting the input randomly or taking some other random action On the same input, a randomized algorithm has multiple possible executions No one input elicits worst-case behavior Typically we analyze the ave ...
... assumption that inputs exhibit some distribution, make your own input distribution by, say, permuting the input randomly or taking some other random action On the same input, a randomized algorithm has multiple possible executions No one input elicits worst-case behavior Typically we analyze the ave ...
Assignment2
... win the first game, the probability you also win the second is 0.2. If you lose the first game, the probability that you win the second is 0.3 (a) Are the two games independent? Explain your answer using probabilities. (b) What’s the probability you lose both games? (.42) (c) What’s the probability ...
... win the first game, the probability you also win the second is 0.2. If you lose the first game, the probability that you win the second is 0.3 (a) Are the two games independent? Explain your answer using probabilities. (b) What’s the probability you lose both games? (.42) (c) What’s the probability ...
CSC384: Intro to Artificial Intelligence Reasoning under Uncertainty
... restricting one’s attention to the events in A. ● We now consider A to be the whole set of events (a new universe of events): Pr(A|A) = 1. ● Then we assign all other sets a probability by taking the probability mass that “lives” in A (Pr(B∧A)), and normalizing it to the range [0,1] by dividing by Pr ...
... restricting one’s attention to the events in A. ● We now consider A to be the whole set of events (a new universe of events): Pr(A|A) = 1. ● Then we assign all other sets a probability by taking the probability mass that “lives” in A (Pr(B∧A)), and normalizing it to the range [0,1] by dividing by Pr ...
Dependent Events
... 2. Getting 3 red gumballs in a row out of a machine of different colors There is not enough information here to determine the probability but we know that the probability depends on how many red gumballs there are, and how many gumballs there are total. ...
... 2. Getting 3 red gumballs in a row out of a machine of different colors There is not enough information here to determine the probability but we know that the probability depends on how many red gumballs there are, and how many gumballs there are total. ...
Menu-Dependent Stochastic Feasibility
... When feasibility is driven by stock-outs, correlation provides a natural way to discuss substitutability/complementarity.3 For example, we say two alternatives are substitutes if there is negative correlation in feasibility because one alternative is less likely to be available in the presence of th ...
... When feasibility is driven by stock-outs, correlation provides a natural way to discuss substitutability/complementarity.3 For example, we say two alternatives are substitutes if there is negative correlation in feasibility because one alternative is less likely to be available in the presence of th ...
Activity: Determining if a Die is Fair
... histogram tell us how many times a given number of heads occurred. The solid circles (connected by a solid line only as a guide to the eye) are the expected values given that the probability of a head is 1/2; this is just the binomial distribution shown in Fig. A1. The data have ...
... histogram tell us how many times a given number of heads occurred. The solid circles (connected by a solid line only as a guide to the eye) are the expected values given that the probability of a head is 1/2; this is just the binomial distribution shown in Fig. A1. The data have ...
Lower Bounds on Learning Random Structures with
... Trakhtenbrot and Barzdin [13] consider the problem of learning arbitrary finite automata using experiments – in each experiment the learning algorithm selects an input string to query and receives the sequence of output symbols generated by the target machine on reading that input. For a DFA, this i ...
... Trakhtenbrot and Barzdin [13] consider the problem of learning arbitrary finite automata using experiments – in each experiment the learning algorithm selects an input string to query and receives the sequence of output symbols generated by the target machine on reading that input. For a DFA, this i ...
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.