Probability Binomial event

There are many other distributions used in statistics besides the normal distribution. Common ones are the yl and the F-distributions (see later) and the binomial distribution. The binomial distribution involves binomial events, i.e. events for which there are only two possible outcomes (yes/no, success/failure). The binomial distribution is skewed to the right, and is characterised by two parameters n, the number of individuals in the sample (or repetitions of a trial), and n, the true probability of success for each individual or trial. The mean is n n and the variance is nn(l-n). The binomial test, based on the binomial distribution, can be used to make inferences about probabilities. If we toss a true coin a iarge number of times we expect the coin to faii heads up on 50% of the tosses. Suppose we toss the coin 10 times and get 7 heads, does this mean that the coin is biased. From a binomiai tabie we can find that P(x=7)=0.117 for n=10 and n=0.5. Since 0.117>0.05 (P=0.05 is the commoniy... 

P is the probability of the binomial event n is the total number of observations r is the number of desired outcome events p is the probability of the desired event occurring one time is the factorial of the value... 

Table 12 illustrates the computation procedure in the case of m = 5 the plant may be envisaged, as in Section V.l, to consist of m cell banks, the quantity j denoting the number of banks switched back into operation. In the specific case of X = p (equi -probability of switching into either direction), Eq. (47) reduces to the binomial probability distribution of selecting j elements out of m identical elements with a single-event probability of xh. 

For further information see Reference 18.] The event might be the presence of any particular attribute in a sample, such as the detection of a pesticide. Only two levels of the attribute are possible, present or not present. If many attributes contribute to the result of an observation, the binomial probability distribution approaida.es a limiting curve whose equation is given by y = (1/ /211) exp[-(2 jx) As... 

Binomial (or Bernoulli) Distribution. This distribution applies when we are concerned with the number of times an event A occurs in n independent trials of an experiment, subject to two mutually exclusive outcomes A or B. (Note The descriptor independent indicates that the outcome of one trial has no effect on the outcome of any other trial.) In each trial, we assume that outcome A has a probability P(A) = p, such that q, the probability of outcome A not occurring, equals (1 - q). Assuming that the experiment is carried out n times, we can consider the random variable X as the number of times that outcome A takes place. X takes on values 1, 2, S,---, n. Considering the event X = x (meaning that A occurs in X of the n performances of the experiment), all of the outcomes A occur x times, whereas all the outcomes B occur (n - x) times. The probability P(X = x) of the event X = x can be written as ... 

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. 

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... 

The Poisson distribution applies to events whose probability of occurrence is small and constant. It can be derived from the binomial distribution by letting... 

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. 

The theoretical model simulates the reaction scheme of the intermittent propagation of Fig. 7 on the basis of a statistical distribution of the polymerization activity onto all molecules (C ) present in the reactor. In other words, the possibility to become an active species is again distributed newly after each insertion step, because the concentration of the different alkyl chains is changed after each insertion step. Figure 14 shows the binomial distribution formula or, more precisely, the Bernoulli scheme for two incompatible events. In this formula, a is the probabiUty for the event, 1—a the non-probability for the event, and v the number of times that the event occurs. 

Modelling of the failure type of class (2) requires one to determine the expected frequency of the shock events and the corresponding conditional probabilities of component failures caused by them. The binomial failure rate model (BFR) is the best known model of this class. For its application observed CCF events are used to calculate the parameter of the binomial distribution [u in Eq. (9.36)]. This then enables one to determine the probabilities of failure combinations (e.g. three-out-of four redundant components) including for combinations which have not been observed. 

One of the best known parametric models is the Binomial Failure Rate (BFR) model (Vesely 1977). The model postulates a shock causal mechanism for the failures. In particular, the model assumes that common cause events occur to the system as a result of shocks that may lead to common cause events of various multiplicities. These shocks hit the system according to a Poisson distribution with rate /x. A fundamental assumption of the BFR model is that, at the occurrence of a shock, all components fail independently with the same probability p. Therefore, the probability 4>k/m that exactly k components fail, whereas m — k... 

For frequent hitter analysis, we defined a frequent hitter score that depends on the number of screens in which a compound participated and on the number of screens where this compound was a hit We aimed at identilying a simple, empirical score that allows us to rank compounds with respect to their promiscuity, also in cases where compounds where tested in a different number of assays. A biological assay system is modeled as a biased coin that yields hit or non-hit with certain probabilities and the various assays to which a compound is subjected as a sequence of independent coin flips. Thus, we use a binomial distribution function to estimate the relative probabiUty of identifying a compound as a hit n times in k independent assays by chance. The probabiUties for the events hit and non-hit were estimated empirically from a set of assays. 

The events that affect the state of the system in each period are as follows First, the agent chooses a one-dimensional service level decision at G where Cl is assumed to be the closed interval [g, d]. The agent incurs a cost of service g at)Qt-i, where the cost function p(-) is strictly increasing and convex. Service level affects stochastically the number of customers who will experience a minor service failure, denoted Ht, as well as the number of major service failures in the same period, denoted Dt. The two random variables Ht and Dt are observed next. The probability distribution for Ht is assumed to be Binomial with number of trials Qt-i and probability of minor service failure for each trial given by a proportional hazards function PH ( t) = j where Oh > 0 ph o) denotes the per-period... 

In situations where there are only two possible outcomes for an event—such as a male/female or yes/no situation—the distribution is said to be binomial Observing a safety valve a number of times and determining the probability that it will be open or closed a specific number of times meets the binomial probability since there are only two possible outcomes, open or closed. The formula presented below can be used to determine the binomial probability (Hays 1988) ... 

Binomial Probability a probability in which only two possible outcomes for an event are possible—such as a heads/tails or yes/no situation the distribution of these probabilities is considered to be binomial. 

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 ... 

Some probability distribution functions occur frequently in nature, and have simple mathematical expressions. Two of the most useful ones are the binomial and multinomial distribution functions. These will be the basis for our development of the concept of entropy in Chapter 2. The binomial distribution describes processes in which each independent elementary event has two mutually exclusive outcomes such as heads/tails, yes/no, up/down, or occu-pied/vacant. Independent trials with two such possible outcomes are called Bernoulli trials. Let s label the two possible outcomes 9 and J. Let the probability of be p. Then the probability of J is 1 - p. We choose composite events that are pairs of Bernoulli trials. The probability of followed by is P0j = p(l - p). The probabilities of the four possible composite events are... 

In Example 1.7 we defined composite events as pairs of elementary events. More generally, a composite event is a sequence of AT repetitions of independent elementary events. The probability of a specific sequence of n s and N n J s is given by Equation (1.7) (page 4). What is the probability that a series of N trials has n s and N - n J s in any order Equation (1.19) (page 12) gives the total number of sequences that have n s and N - n J s. The product of Equations (1.7) and (1.19) gives the probability of n 8 and N - n s irrespective of their sequence. This is the binomial distribution ... 

The multinomial probability distribution is a generalization of the binomial probability distribution. A binomial distribution describes two-outcome events such as coin flips. A multinomial probability distribution applies to t-outcome events where n, is the number of times that outcome i = 1, 2,3,..., f appears. For example, t = 6 for die rolls. For the multinomial distribution, the number of distinguishable outcomes is given by Equation (1.18) W = AH/lniln in j nj ). The multinomial probability distribution is... 

Binomial Distribution (BD) The BD describes a wide variety of situations in which the probability of the ontcome of a certain event, q, remains fixed for all trials for example, the probability of rolling a certain number of a die. 

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 Poisson distribution is a special case of a binomial distribution, if the probability of failure p or probability of successes q is very small and the number of experiment n is very large. The conditions under which the Poisson distribution holds are the counts of rare events, all events are independent and the average rate does not change over the period of interest. The Poisson probability function is given as follows ... 

The event tree approach (Fig. 4) is used to track outcomes from gas leakage and branch outward based on binomial choices on possible outcomes (e.g., ignition yes/no). Using the equation (1), (2), (3) can determine the final probabilities of every estimated outcomes. 

The Poisson distribution is another distribution for counts. Specifically the Poisson is a distribution which counts the number of occurrences of rare events. Unlike the binomial which counts the number of events (successes) in a known number of independent trials, the number of trials in the Poisson is so large that it isn t known. However looking at the binomial gives us way to start our investigation of the Poisson. Let y be a binomial random variable where n is very large, and tt is very small. The binomial probability function is... 

The probability that the event will occur X times in iV trials is given by the binomial distribution (Fig. 2.5)... 

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