An Interview with Ann Varela: Jacques Bernoulli

Feb 20, 2018 by

Michael F. Shaughnessy –

1) Ann, in this interview, we are going to do something radically different. I have been so overwhelmed studying Bernoulli that I have decided to simply list everything that he is known for- and associated with–and you can briefly tell us about each of these things- and their relevance to math!

The Bernoulli family produced eight distinguished mathematicians, which is the most in history. The family fled the Netherlands in 1583 for Basel, Switzerland because of increasing taxation and religious persecution. Jacques, also known as James and Jakob, lived from 1654-1705 and was the eldest son of Nicolaus Bernoulli. The following topics are a sampling of the Jacques Bernoulli legacy.

Bernoulli Trial

In probability theory, the Bernoulli trial is a random experiment with exactly two possible outcomes, “success” or “failure”. The probability of success is the same each time the experiment is conducted. The term “success” means the indicated conditions have been met. The term “failure” means the indicated conditions have not been met. Common Bernoulli trials involve a coin toss, a die roll, and opinion poll responses in a “yes/no” format.

With the coin toss experiment, one may observe either “heads” or “tails”; thus, two possible outcomes are feasible. “Heads” is traditionally deemed as success; therefore, “tails” would signify failure.

For the die roll experiment, there are six possible outcomes for the standard die, numbers one through six. Since each number appears only once on the six-sided die, the likelihood of any one number is equally likely to occur. That is, each number from one to six has a one in six chance of occurring. Therefore, the probability for each outcome is 1/6. Suppose “four” is defined as “success”, then one out of six describes the probability of success. It follows that the other five outcomes out of six possible outcomes, 5/6, would represent “failure”.

Bernoulli Process

A Bernoulli process is a sequence of Bernoulli trials possessing the following characteristics:

  • The trials are independent of each other. This means the outcome of one trial does not affect the outcome of subsequent trials.
  • There are only two possible outcomes for each trial, labeled S (success—the event occurs) or F (failure—the event does not occur). Success generally takes on the discrete value of 1, whereas failure takes on the discrete value of 0.
  • The probability of success is the same for each trial. This probability is denoted by p, and probability of failure is denoted by q in the equation q = 1 – p.

One of the most common examples of the Bernoulli process is the coin toss experiment. Before the experiment begins, the experimenter chooses one of the coin’s sides to represent “success”. For example, “heads” may represent a success. The coin is then flipped x number of times and the results tallied. One could also draw a tree diagram representing possible outcomes of tossing the coin two times as seen in Figure 1.

Figure 1. Tree Diagram for Coin Flipping Outcomes.

Rolling a die is another familiar example of the Bernoulli process. This time, assuming a standard die is used, a number from one to six is chosen at random to represent success. The die is then rolled x number of times and the results tallied.

Bernoulli Random Variable

In statistics, a random variable is a variable whose possible values are the outcomes of a random event. The random variable must be measureable. In other words, a real number is associated with each outcome. In the coin toss experiment, the random variable x is associated with the real number 0 when the outcome is “failure”. If the outcome is “success”, x takes on the value of 1.

Once the probability of the random variable is calculated, the random variable may be displayed in a probability distribution.

Bernoulli Scheme

A Bernoulli scheme is similar to the Bernoulli process, but it has more than two possible outcomes. In contrast, it has a sequence of random variables, each of which may assume one of N distinct values with the same probability distribution. As seen in Figure 2, the summation of the probability distribution has outcome i occurring with probability pi and i = 1, …,N. The sum of the probability distribution must equal one.

Figure 2. Summation of Probabilities.

The Bernoulli scheme is used to analyze dynamical systems in which a function describes the relationship between time and a point in space.

Bernoulli Sampling

Bernoulli sampling is used for finite population sampling. A population is considered to be finite if it is possible to count its individuals. For example, the number of ice cubes in a tray, the number of pencils on a desk, or the number of passengers on a train, are finite populations. Notice how population does not necessarily pertain to people.

Bernoulli sampling is a sampling process where each element of the population is subjected to an independent Bernoulli trial, which determines whether the element becomes part of the sample. It is imperative that all elements of the population have equal probability of being included in the sample. The sample size is not fixed and may assume a value between zero and N.

Bernoulli Distribution

The coin toss experiment previously discussed in the Bernoulli trial may be used to illustrate the Bernoulli distribution. Suppose a coin is tossed a total of ten times and that the coin landed on “heads” three times. Since it has already been established that “heads” represents “success”, Figure 3 shows the probability distribution table for the discrete random variable x:

Figure 3. Probability Distribution Table.

Now that the probabilities for the experiment’s outcomes are known, the Bernoulli distribution may be constructed. The distribution shows the probability associated with each possible outcome of the coin toss experiment. See Figure 4.

Figure 4. Bernoulli Distribution.

Bernoulli’s Golden Theorem

Presently, Bernoulli’s Golden Theorem is commonly known as the law of large numbers (LLN). Back in 1713, Jacque Bernoulli spent over twenty years developing a mathematical proof for the law of large numbers for binary random variables.

The law of large numbers is used in probability theory to describe the result of performing the same experiment for a large number of trials. The LLN states that the average of the outcomes obtained from a large number of trials should be close to the expected value (sample mean), and will tend to become closer as more trials are executed. It may be obvious, but it is important to reiterate that the law of large numbers applies only when a large number of observations is considered.

When considering the fair coin toss experiment, the only possible outcomes are “heads” or “tails”, so the probability of tossing “heads” is ½ or 0.5, which is exactly the same as the probability of tossing “tails”. Now, imagine a coin tossed several times lands on “heads” each time and perhaps two more tosses result in “tails” for those outcomes, the law of large numbers shows how the number of “heads” and “tails” will balance out if a large number of tosses are observed. Therefore, if one is planning a trip to Las Vegas or Atlantic City, the old adage of, “Quit while you’re ahead” may make more sense now because a winning streak will eventually produce “not winning” the more one plays a given game of chance.

Hidden Bernoulli Model

The hidden Bernoulli model is a form of probability evaluation that is assumed to be a Bernoulli model with unobserved (hidden) states. In the field of information technology, a program is said to be stateful if it is designed to remember preceding events or user interactions. The information that is remembered is called the state of the system. The hidden Bernoulli model is an example of a stochastic model for a process that has some kind of uncertainty. The hidden Bernoulli model has been used for modeling acoustic-unit duration (or voice recognition), which is a phoneme recognition task.

Bernoulli Polynomial

Many specialized functions are rooted in Bernoulli polynomials. Two common specialized functions are the Riemann zeta function and the Hurwitz zeta function. What is noteworthy about these two functions is that the number of intersections with the x-axis in the unit interval does not go up as the degree (power) of the polynomials goes up, as is typical in other types of polynomials. In fact, the Bernoulli polynomials may approach the sine and cosine functions, as seen in Figure 5.

The unit interval is a subset of the real number line. As a closed set is defined as the set of all real numbers that are greater than or equal to zero and less than or equal to one. Using interval notation, the closed unit interval is written as [0, 1].

Figure 5. Bernoulli Polynomials.

Bernoulli’s Inequality

Bernoulli’s inequality is used in real analysis, which is the study of the theory of functions of a real variable. Real analysis focuses mainly on analytic properties of the real functions and sequences of real numbers. This particular inequality approximates exponentiations of 1 + x. Exponentiations are mathematical operations written as bn, where b is the base and n is an exponent. If the exponent is a positive integer, repeated multiplication of the base is indicated. The Bernoulli inequality is in the form (1 + x)n > 1 + nx, where x is a real number and x > -1, x cannot equal 0, and n is an integer greater than 1. Bernoulli’s inequality is a form of deductive reasoning, which is frequently used as an inferential argument (proof) of other inequalities.

Bernoulli Differential Equation

What is exciting about the Bernoulli differential equations is that they are nonlinear differential equations with known exact solutions. In 1695, Jacque Bernoulli’s differential equations were of the form: y’ + P(x)y = Q(x)yn, where n is a real number, but not equal to zero or one.

Lemniscate of Bernoulli

The Lemniscate was described in 1694 by Jacque Bernoulli as a variation of an ellipse. Apparently, Bernoulli was not aware of this Lemniscate existing as a special case of the Cassini oval, which was previously described by Cassini in 1680. Characteristics of the Lemniscate include a plane curve consisting of two given points F1 and F2, known as foci, at a distance 2a from each other, and the locus of points for which the product of the distances to each of two fixed focal points is a constant, as shown in Figure 6. The length along the Lemniscate curve (arc length) was studied by Gauss and Euler. Their research ultimately provided the preliminary groundwork for later work on elliptical functions.

Figure 6. Lemniscate of Bernoulli.

Bernoulli Numbers

In the branch of mathematics called number theory, the Bernoulli numbers are a sequence of rational numbers. Bernoulli wrote an expression for each of these sums, with coefficients involving the Bernoulli numbers. See Figure 7.

Figure 7. Summae Potestatum.

Bernoulli’s Formula

The Bernoulli formula, also known as Faulhaber’s formula, is used to calculate the sum of powers of consecutive positive integers. Figure 8 shows the formula when p represents a power (exponential value), n denotes the last positive integer term in the sequence, and the summation symbol to the left of the equation indicates the initial power value and final term to include in the sequence.

Figure 8. Bernoulli Numbers.

Bernoulli Map

The Bernoulli map is a recurrence relation. Recurrence relations are equations that recursively define a sequence based on a rule that gives the next term as a function of the previous term(s). A recurrence relation can be regarded as determining a discrete dynamical system, which are concerned with the behavior of a function over time, such as the Bernoulli map.

The Bernoulli map is also known as the iterated function map when an initial input value is used in the function and this process is recurrent. In addition, it is seen as a bit shift map when the value of an iterate is written in binary notation, as the next iterate is obtained by shifting the binary point one bit to the right, and if the bit to the left of the new binary point is a “one”, replacing it with a zero.

Dyadic map is yet another name for the Bernoulli map. The dyadic map is modeled in the theory of deterministic chaos. In chaos theory, the behavior of dynamical systems is studied. Observations within the outward randomness of chaotic complex systems reveal that there are fundamental patterns or repetitions present. The future behavior of these systems is determined by their initial conditions.

Bernoulli Operator

The Bernoulli operator is a transfer operator that encodes information regarding an iterated map, as mentioned above. This operator is commonly used to study the movement or behavior of dynamical systems, such as water flow in a pipe, a swinging pendulum, as well as fractals. The transfer operator of the Bernoulli map is precisely solvable.

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