Binomial distribution, discrete probability

The binomial distribution describes a population whose members have only certain, discrete values. A good example of a population obeying the binomial distribution is the sampling of homogeneous materials. As shown in Example 4.10, the binomial distribution can be used to calculate the probability of finding a particular isotope in a molecule. 

Binomial distribution if an event has a probability p of occurring in one trial, the binomial distribution gives its probability of occurring r times in n trials. The parameter r has only discrete whole number values, and the value of p at each r is given by... 

The binomial distribution predicts the probability of observing any given number (k) of successes in a series of n random independent trials. This distribution can only be applied to discrete population sizes. In its simplest form, the outcome of a trial can only be one of two events, yes or no, success or failure, but the analysis can be extended to situations where more outcomes are possible. 

Binomial distribution. This is a discrete distribution in finite space The probability that the random variable n takes any integer value between 0 and N is given by... 

The Poisson distribution describes the probability of a discrete number of events occurring within a fixed interval, given that the probability of the event occurring is independent of the size of the interval. For example, suppose that a manufacturing defect occurs with an average rate of occurrence p and the products are manufactured over an interval n. The expected number of defects is clearly np. The Poisson distribution is one way of describing the probability that x defects will occur in an interval n. This distribution is a generalization of the binomial distribution to an infinite number of trials. The mathematical form of the Poisson distribution [2] is... 

Discrete probability distributions include the Binomial distribution and the Poisson distribution. 

We proceed by introducing a free-volume unit p and discrete free-volume parcels of size fj = jP for 0< j < n. For a binomial distribution with two parameters, unit probability pr, and number of states n +1, we have... 

D.5 DISCRETE PROBABILITY DISTRIBUTIONS D.5.1 Binomial and Multinomial Distributions... 

The binomial distribution gives the discrete probability distribution of obtaining n successes out of N Bernoulli trials. The result of each Bernoulli trial is true with probability p and false with probability q = I- p (Figure D.l). 

Binomial Distribution A distribution of data or results describing probabilities of the outcome of trials that can have one or two mutually exclusive results (e.g., exposure above or below a permissible exposure limit or PEL ). This theoretically discrete probability distribution for a binomial random variable is represented as ... 

The binomial distribution, denoted by 8 ( , <7), is a discrete distribution used to model the outcome of a series of binary (0 and 1 or yes or no) events. For each trial or realisation, the value 1 can occur with probability q and the value 0 with probability 1 - <7. It is assumed that there are k trials and the number of 1 s is s. The order in which the events occur is not important, only their total number, for example, 1,0,0, 1 ... 

Table 5. Binomial Distribution. This is a discrete distribution representing the probabilities of success in N trials for a population or sample in which only two outcomes are possible, but for which the eventual outcome is fixed and known if an infinite number of trials are made.f This eventual outcome is fixed by the conditions, such as 0.5 for one face of a coin or 0.1667 for one face of a six-sided die. Values in the body of the table represent the cumulative piobabihty of X or more successes in N trials. In applications to acceptance or attribute sampling, the table gives the probability of X or more acceptances (or rejections) in a single sample of N items. In either case the known or fixed probability of the result (success or failure) for the entire population is represented by p.

Gaussian and Poisson distributions are related in that they are extreme forms of the Binomial distribution. The binomial distribution describes the probability distribution for any number of discrete trials. A Gaussian distribution is therefore used when the probability of an event is large (this results in more symmetric bell-shaped curves), whereas a Poisson distribution is used when the probability is small (this results in asymmetric curves). The Lorentzian distribution represents... 

The Binomial distribution, also known as Bernoulli distribution, belongs to the discrete distribution. The specific feature of the binomial distribution is that it can accept only 2 values. This means that a component failure is present x = 0 or is not present X= 1. The probability of exactly x successes is given by this formula ... 

The binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial when n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is n x repeated Bernoulli trial. The binomial distribution is the basis for the popular binomial test of statistical significance. 

In order to determine a for uncorrelated data, we only need to calculate the probability PQii) that in the rth thread a state with energy E (for simphcity we assume that the problem is discrete) is hit /j, times in M, trials, where each hit occurs with the probability hit This leads to the binomial distribution with the hit average hi) = Mjphit- In the limit of small hit probabilities (a reasonable assumption in general if the number of energy bins is large, and, in particular, for the tails of the histogram), the binomial turns into the Poisson distribution P hi) /hi with identical variance and expectation value, a, = hi). Insertion... 

It is helpful to have standard probabihty models that are useful for analyzing large biological data, in particular bioinformatics. There are six standard distributions for discrete r.v. s, that is, BemouUi for binary r.v. s, (e.g., success or failure), binomial for the number of successes in n independent BemouUi trials with a common success probabihty p, uniform for model simations where aU integer outcomes have the same probabihty over an interval [a, b, geometric for the number of trials required to obtain the first success in a sequence of independent BemouUi trials with a common success probabihty p, Poisson used to model the number of occurrences of rare events, and negative binomial for the number of successes in a fixed number of Bemoulh trials, each with a probability p of success. 

Table 6. Poisson Distribution. This is a discrete distribution approximating the binomial when the total number of items of data (the populations) is very large, but the probability (p) is very small and the sample is small compared with the population...

The binomial probability distribution is an important example of a discrete distribution. Say that we toss an unbiased coin n times and want to find the probability that heads will come up m times. For a single toss, the probability of heads is equal to 1 /2. The probability that heads will come up every time on m consecutive throws is... 

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