Combinatorics
... Solution. We can imagine filling in five blanks ___ ___ ___ ___ ___, left to right, to create the (equally likely) outcomes. The total number of these is 5! = 120, by the Product Rule. To count the number of these satisfying the given condition, simply adapt the Product Rule accordingly: Filling in ...
... Solution. We can imagine filling in five blanks ___ ___ ___ ___ ___, left to right, to create the (equally likely) outcomes. The total number of these is 5! = 120, by the Product Rule. To count the number of these satisfying the given condition, simply adapt the Product Rule accordingly: Filling in ...
Random Walks in Swarm Robotics: An Experiment with Kilobots
... is terminated after a fixed amount of time T or when a sufficient number of robots have discovered the target. We consider two simple scenarios that represent two common situations in swarm robotics. The first scenario consists in a bounded space surrounded by walls (square area with side L) in which ro ...
... is terminated after a fixed amount of time T or when a sufficient number of robots have discovered the target. We consider two simple scenarios that represent two common situations in swarm robotics. The first scenario consists in a bounded space surrounded by walls (square area with side L) in which ro ...
Introduction to The Central Limit Theorem
... Rule 3. When the population distribution is normal, the sampling distribution of Y is also normal for any sample size n. Rule 4. (Central Limit Theorem) When n is sufficiently large, the sampling distribution of Y is well approximated by a normal curve, even when the population distribution is not i ...
... Rule 3. When the population distribution is normal, the sampling distribution of Y is also normal for any sample size n. Rule 4. (Central Limit Theorem) When n is sufficiently large, the sampling distribution of Y is well approximated by a normal curve, even when the population distribution is not i ...
Normal Random Variables
... Going Backwards: Sometimes the question will be asked in reverse. In other words you will be given a probability or a percentage and asked to find the range of X values. For these questions we will use the “invnorm” function, found by pressing 2nd and DISTR, on our calculator which finds the x value ...
... Going Backwards: Sometimes the question will be asked in reverse. In other words you will be given a probability or a percentage and asked to find the range of X values. For these questions we will use the “invnorm” function, found by pressing 2nd and DISTR, on our calculator which finds the x value ...
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.