A Tail Bound for Read-k Families of Functions
... when Y1 , . . . , Yr form a martingale, in which case Azuma inequality and its generalizations give bounds which are comparable to Chernoff bound. We consider in this paper another model of weak dependence. Assume that the variables Y1 , . . . , Yr can be factored as functions of independent random ...
... when Y1 , . . . , Yr form a martingale, in which case Azuma inequality and its generalizations give bounds which are comparable to Chernoff bound. We consider in this paper another model of weak dependence. Assume that the variables Y1 , . . . , Yr can be factored as functions of independent random ...
Notes for Math 450 Lecture Notes 3
... We saw in Lecture Notes 2 that if random variables X and Y are independent and have probability density functions fX and fY , then the sum X + Y has probability density function equal to the convolution fX ∗ fY . Therefore, one way to observe the convergence to the normal distribution claimed in the ...
... We saw in Lecture Notes 2 that if random variables X and Y are independent and have probability density functions fX and fY , then the sum X + Y has probability density function equal to the convolution fX ∗ fY . Therefore, one way to observe the convergence to the normal distribution claimed in the ...
PDF
... A real random variable X defined on a probability space (Ω, F, P ) is said to be stable if 1. X is not trivial; that is, the range of the distribution function of X strictly includes {0, 1}, and 2. given any positive integer n and X1 , . . . , Xn random variables, iid as X: t ...
... A real random variable X defined on a probability space (Ω, F, P ) is said to be stable if 1. X is not trivial; that is, the range of the distribution function of X strictly includes {0, 1}, and 2. given any positive integer n and X1 , . . . , Xn random variables, iid as X: t ...
Chapter 5. Conditional expectations and conditional probabilities
... now is that in most books the notation E(X|Y ) is used instead of the notation (5.1.6). This is not very good, because it could lead to confusing both versions. Anyway, I am going to follow the notation (5.1.6) that is used in some books. Of course, there will also be defined the conditional probabil ...
... now is that in most books the notation E(X|Y ) is used instead of the notation (5.1.6). This is not very good, because it could lead to confusing both versions. Anyway, I am going to follow the notation (5.1.6) that is used in some books. Of course, there will also be defined the conditional probabil ...
Elementary Statistics 12e
... A conditional probability of an event is a probability obtained with the additional information that some other event has already occurred. P( B | A) denotes the conditional probability of event B occurring, given that event A has already occurred, and it can be found by dividing the probability of ...
... A conditional probability of an event is a probability obtained with the additional information that some other event has already occurred. P( B | A) denotes the conditional probability of event B occurring, given that event A has already occurred, and it can be found by dividing the probability of ...
Chapter 6
... “success” is one of these three outcomes, p(bothheads) = 1/3. Ans: The probabilities of the three outcomes are not equal. Using {HH, HT, TH, TT} as the sample space, the correct answer, 1/4, is obtained. 12. Let A be the set of all strings of decimal digits of length 5. For example 00312 and 19483 a ...
... “success” is one of these three outcomes, p(bothheads) = 1/3. Ans: The probabilities of the three outcomes are not equal. Using {HH, HT, TH, TT} as the sample space, the correct answer, 1/4, is obtained. 12. Let A be the set of all strings of decimal digits of length 5. For example 00312 and 19483 a ...
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.