Bernoulli experiment

Bernoulli and Euler dominated the mechanics of flexible and elastic bodies for many years. They also investigated the flow of fluids. In particular, they wanted to know about the relationship between the speed at which blood flows and its pressure. Bernoulli experimented by puncturing the wall of a pipe with a small, open-ended straw, and noted that as the fluid passed through the tube the height to which the fluid rose up the straw was related to fluid s pressure. Soon physicians all over Europe were measuring patients blood pressure by sticking pointed-ended glass tubes directly into their arteries. (It was not until 1896 that an Italian doctor discovered a less painful method that is still in widespread... 

Following Durovic and Kovacevic (1995), let us now consider a measurement sample recorder y(,), i = 1,..., M (here, i can be considered as each sampling instance), from a distribution F(y), corresponding to a probability density function f(y). When the samples >>(, ) are rearranged in ascending order y,, the probability that an observation y will have rank i in the ordered sequence (y,-) follows from the Bernoulli experiment (Papoulis, 1991) ... 

We model the binder by a so-called BemouUifilUng, where in each layer Bernoulli experiments with probability p > 0 are performed for each cell of the PLT independently. If the Bernoulli experiment is successful, the cell is filled with binder, either completely or partially. An example can be seen in Figure 24.5, where in (a) the fiber system without binder is shown, (b) depicts the fiber system with a cell completely filled with binder, and in (c) the fiber system with a partially filled cell can be seen. A 3D realization of the stochastic multi-layer model including binder is shown in Figure 24.6. 

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. 

Nature Consider an experiment in which each outcome is classified into one of two categories, one of which will be defined as a success and the other as a failure. Given that the probability of success p is constant from trial to trial, then the probabinty of obseivdng a specified number of successes x in n trials is defined by the binomial distribution. The sequence of outcomes is called a Bernoulli process, Nomenclature n = total number of trials X = number of successes in n trials p = probability of obseivdng a success on any one trial p = x/n, the proportion of successes in n triails Probability Law... 

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 ... 

The Bernoulli distribution applies wherever there are just two possible outcomes for a single experiment. It applies when a manufactured product is acceptable or defective when a heater is on or off when an inspection reveals a defect or does not. The Bernoulli distribution is often represented by 1 and 0 as the two possible out-... 

A discrete distribution function assigns probabilities to several separate outcomes of an experiment. By this law, the total probability equal to number one is distributed to individual random variable values. A random variable is fully defined when its probability distribution is given. The probability distribution of a discrete random variable shows probabilities of obtaining discrete-interrupted random variable values. It is a step function where the probability changes only at discrete values of the random variable. The Bernoulli distribution assigns probability to two discrete outcomes (heads or tails on or off 1 or 0, etc.). Hence it is a discrete distribution. 

A more useful and more frequently used distribution is the binomial distribution. The binomial distribution is a generalization of the Bernoulli distribution. Suppose we perform a Bernoulli-type experiment a finite number of times. In each trial, there are only two possible outcomes, and the outcome of any trial is independent of the other trials. The binomial distribution gives the probability of k identical outcomes occurring in n trials, where any one of the k outcomes has the probability p of occurring in any one (Bernoulli) trial ... 

In the earlier days of the petroleum age, many pipe experiments were conducted. In the quest for the magic formula, one was found to be the closest to utopia even to this day, called the Darcy formula. The Darcy formula is derived manually from the Bernoulli principle, which simply describes the energy balance between two points of a fluid flowing in a pipe. This energy equation is also applicable to a static condition of no flow between the two points. The classic Bernoulli energy equation [1] is ... 

Streamlines are hypothetical lines without width drawn parallel to all points to the motion of the fluid. As velocity increases, pressure decreases. Pressure field around an object is the reverse of velocity field. This may appear to contradict common experience. However, it follows from the principle of conservation of energy and finds expression in Bernoulli s theorem. 

The geometric distribution indicates the probability of conducting x trials to obtain a success in an experiment in which there are only two possible outcomes. Like the binomial distribution, this is another Bernoulli process. Each trial is assumed to be independent, and the probability of observing a success is constant over all trials, denoted p. The probability distribution for the geometric distribution [2] is... 

Colloidal particles are large enough that any given particle, not in the center of the tube, will experience a pressure difference across its diameter due to the higher velocity on one side than on the other. Thus, the particle will experience a "lift-force" tending to move it away from the wall toward the center of the channel. This "lift-force" is not unlike the aerodynamic lift on an airfoil due to the "Bernoulli effect". 

Steve Czamecki from Owego, New York, was the first person to provide an interesting explanation for this mystery. To understand his argument, we define a Bernoulli trial as a random experiment with only two possible outcomes. The person sliding over a hole is a Bernoulli trial the individual will either drop or pass over the hole with probability p and (1 -p), respectively. Therefore, to make it all the way to the bottom, the person must achieve the pass over result of all the individual Bernoulli trials. This probability is given by P = (1 - p) °, which is 1/1,024 when p is 1/2. 

Czamecki next asks us to forget about the details of the slide itself and instead only observe the people climbing up the ladder to the slide, and also observe whether or not they appear at the bottom (after an appropriate time interval). This means we are observing another random experiment with two possible outcomes Either the person makes it or does not make it to the bottom. In other words, every time a person tries the slide, it is a Bernoulli trial with outcomes made it to the bottom or did not make it to the bottom. ... 

Bernoulli Distribution Ar.v. that can take only two values, say 0 and 1, is called a Bernoulli r.v. The Bernoulli distribution is a useful model for dichotomous outcomes. An experiment with a dichotomous outcome is called a Bernoulli trial. 

Example 2.16 Bernoulli Trials. Many experiments can be modeled as a sequence of Bernoulli trials, the simplest being repeated tossing of a coin p = probability of a head, X = 1 if the coin shows a head. Other examples include gambling games (e.g., in roulette let Z = 1 if red occurs, so p = probability of red), election polls (Z = 1 if candidate A gets a vote), and incidence of a disease (p = probabiUty that a random person gets infected). Suppose Z BemouUi(0.7). P(X = 1) = 0.7. The R code to compute P(X = 1) is as follows ... 

Binomial Distribution Some experiments can be viewed as a sequence of independent and identically distributed (i.i.d.) Bernoulli trials, where each outcome is a success or a Tailure. The total number of successes from such an experiment is... 

Poisson Distribution The Poisson distribution can serve as a model for a number of different types of experiments. For example, when the number of opportunities for the event is very large but the probability that the event occurs in any specific instance is very small, the number of occurrences of a rare event can sometimes be modeled by the Poisson distribution. Moreover, the occurrences are i.i.d. Bernoulli trials. Other examples are the number of earthquakes and the number of leukemia cases. 

OC curves for standard acceptance-sampling plans are derived under the assumption that the quality of items can be modeled as independent and identically distributed (i.i.d.) Bernoulli random variables. Although this model is often plausible, the quality of items produced by some processes exhibit statistical dependence. The goal of this simulation experiment is to estimate the OC curve for sampling plan (10, 1) when item quality is dependent. 

Bernoulli triai An experiment in which there are two possible independent outcomes, for example, tossing a coin. It is named after the Swiss mathematician Jakob... 

Stephen Hales, an English clergyman and physicist, carried out a classic experiment in 1732 to determine blood pressure. He connected a U tube to the carotid artery of a mare and observed the height that blood rose in the tube. Then, using fluid dynamic principles, he calculated the velocity of blood in the aorta, force of contraction, and stroke volume. This work has been the foundation of modem hemodynamics and was used by Bernoulli in his quite accurate calculation of cardiac output in 1737. 

Interpretation. Consider a dichotomous experiment that has only two possible outcomes (alternative events) like a coin toss. One of the alternative outcomes (e.g., heads ) is generally called success while the other (e.g., tails ) failure. The probability of success is denoted by p, that of failure by q = I — p. Now, the Bernoulli variable X is defined as follows ... 

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