Bernoulli trial

In the example presented in the previous paragraph, two results of each event have been considered. In this problem, an event has been repeated with two possible outcomes. One of them is usually referred to as the success (p) ahd die other as failure (q = 1 — py These independent events are known as Bernoulli trials, after the Bernoulli brothers The general expression for thti probability is then given by... 

For syndiotactic polypropylene the symmetric Bernoulli trial, expressed in m and r dyads, is quite adequate for the representation of experimental data, and agrees with the stereochemical control being exerted by the growing chain end (145, 409). In its turn, atactic polypropylene is considered as a mixture of the products of two superposed processes, of the type discussed for isotactic and syndiotactic polymers, and is described by a simplified two-state model (145). 

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 ... 

PROBABILITY THEORY A Concise Course, Y.A. Rozanov. Highly readable, self-contained introduction covers combination of events, dependent events, Bernoulli trials, etc. Translation by Richard Silverman. 148pp. 5X x 8X. 

Games of chance inspired several early investigations of the random sum representing the number of successes in n Bernoulli trials ... 

Here the random variable Xj equals 1 if trial j gives a success and 0 if trial j gives a failure the probabilities of these two outcomes for a Bernoulli trial are p and q = 1 — p. respectively. 

Bayes demonstrated his theorem by inferring a posterior distribution for the parameter p of Eq. (4.3-2) from the observed number k of successes in n Bernoulli trials. His distribution formula expresses the probability, given k and n, that p lies somewhere between any two degrees of probability that can be named. The subtlety of the treatment delayed its impact until the middle of the twentieth century, though Gauss (1809) and Laplace (1810) used related methods. Stigler (1982, 1986) gives lucid discussions of Bayes classic paper and its various interpretations by famous statisticians. 

Building on this notation for a single patient, the individual binary responses of a group of n patients administered the given treatment can be thought of as a series of Bernoulli trials and described using the binomial distribution, with... 

The probability Frm (x) that crm < x is identical to the probability that no more than r — 1 measurements out of m result in cr m > x. Every observation is considered as a Bernoulli trial with probabilities of success and failure F(x) and 1 - F x), respectively. Thus... 

In the case of independent, identically distributed variates, each one of the observations is a Bernoulli trial therefore the probability density function of Nx(m) is, in terms of the parent distribution F(jc)... 

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. ... 

We can now interpret the slide problem as an example of a sequence of independent, repeated Bernoulli trials with probability P of success. Here a success occurs when a person makes it to the bottom of the slide. If we let the random variable N represent the number of unsuccessful trials (attempts) required before the first success for that person (that is, success is achieved on trial N+1), then N has a geometric density defined by the equation P N) = P( - P). Here, P N) is the probability that N unsuccessful trials occur before the successful trial. This is the exponential decay law observed in Figure 10.1. Steve Czamecki notes that the expected value of N (the average number of attempts required before the successful trial) will be E N)- -P)/P. For the example with 10 holes, 1,023 unsuccessful attempts are required, on average, before the successful attempt. Thus, our intuition that the person will achieve one success out of 1,024 attempts, on the average, is correct. 

Bernoulli trials. Independent trials of constant probability with binary outcomes. 

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... 

Geometric Distribution The geometric distribution models the number of i.i.d. Bernoulli trials needed to obtain the first success. It is the simplest of the waiting time distributions, that is, the distribution of discrete time to an event, and is a special case of the negative binomial distribution (we will show this distribution in the next section). Here the number of required trials is the r.v. of interest. As an... 

Negative Binomial Distribution The binomial distribution counts the number of successes in a fixed number of Bernoulli trials, each with a probability p of success. Instead, suppose that the number of successes m is fixed in advance, and the random variable X is the number of trials up to and ineluding this /nth success. The random variable X is then said to have the negative binomial distribution. The probability distribution of X is found as follows. 

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. 

Using this assignment, we calculate the fraction of each configuration. Table III shows the fractional area of methine peaks. The best fits of Bernoulli trial, the 1st-Markov trial, and the 2nd Markov trial are also listed. The data at the triad level are also shown here. The data at the traid level are all in agreement with any statistics. However, the data at more higher level are not agreement with any statistics. The... 

Pascal s distribution (negative binomial distribution) The distribution of the number of independent Bernoulli trials performed up to and including the i success. The probability that the number of trials, x, is equal to fcis given by P x=k) = Cr ip q " ... 

From the results of temperature studies it was concluded that a full description of the stereochemical arrangements of the units in the chain is given by Bernoulli trial statistics, requiring a single parameter a, which is sensitive only to the polymerization temperature (Fox and Schnecko, 1962). The quantity a is the probability that a monomer unit will add to give a configuration the same as that of the terminal unit before addition. One can write that... 

Three basic types of Bernoulli distribution classified according to the ratio of p ("success" -O-1) to q ("failure" -O- 0). Any series of Bernoulli trials results in binomial distribution (seeO Fig. 9.3), however only the third type, characterized by a low probability of success, leads to Poisson distribution. Radioactive decay usually belongs to the latter category... 

The repeated independent trials associated with such a dichotomous game are called Bernoulli trials. 

Interpretation. Consider a dichotomous game. Let p denote the probability of success in a single trial. Suppose that a series of n Bernoulli trials have been performed. Let Xi,X2,...,X denote the independent Bernoulli variables belonging to the respective trials. Then the random variable X = Xi- ------- -X has a B(n, p) binomial distribution. 

Note that X means the number of successful outcomes in the series of n Bernoulli trials. Note also that the above interpretation justifies the use of the symbol B 1, p) for the Bernoulli distribution. 

The discrete equivalent of the Poisson process is related to a series of Bernoulli trials, in which case the individual trials (e.g., coin tosses) can be assigned to individual discrete moments (i.e., to the serial number of the toss, n). It is clear that the process X(n) has a B n, p) binomial distribution - the distribution characteristic of the number of heads turning up in a series of n tosses. 

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