Bernoulli process

A Bernoulli process or a Bernoulli chain is a time- discrete stochastic process that consists of a finite or countably infinite sequence of independent trials with Bernoulli distribution, that is, for each of the time points 1, 2, 3, ... is " diced " if an event occurs with probability or not.

Example of a possible realization of a Bernoulli process, with; the symbol ♦ stands for " event occurs " (short "success" ), ◊ for " event does not occur " ( "failure" ):

◊ - ♦ - ◊ - ♦ - ◊ - ◊ - ♦ - ◊ - ♦ - ◊ - ♦ - ◊ - ◊ - ◊ - ◊ - ◊ - ♦ - ◊ - ◊ - ◊ - ◊ - ◊ - ◊ - ◊ - ...

The process can be described by a sequence of independent random variables, one of which, and with the probability assumes a value of each of the constant probability value of 1 ( success ) 0 ( Failure ).

Depending on the problem we are interested for one or more of the following random variables:

• The number of successful attempts by conducting a total trials; it follows a binomial distribution. It is true.
• The number of experiments needed to achieve a given number of successes; it follows the negative binomial distribution. In particular, the waiting time for the first success is geometrically distributed.

Properties

The number of successes after trials in a Bernoulli process is a special Markov chain: In the " time step " of the system according to the probability of the "state " is in the state; otherwise it remains in the state.

The random variable that indicates how many of Bernoulli trials were successful, follows the binomial distribution. We derive this distribution is an example of her with a cube.

Examples

• During the six dice will considered a success; the probability of success, then, is the complementary failure probability. Asked whether now after the probability of throwing in litters exactly sixes. The answer to this question can be found as follows: The probability of only two sixes, then throw three non- sixes is. But since it does not depend on the order, this probability must be multiplied by the number of ways, two ( indistinguishable ) six litters to distribute to five litters. According to the combinatorics this number is " 2 5" given by the binomial coefficients; the required probability is:

• A drunken pedestrian (or a diffusing particle) moves on a line at each step with probability forward, with the probability backwards. We want to know, for example, the distance from the starting point. Such a model is in physics as one-dimensional random walk ( random walk ) called. The position of the pedestrian steps can be represented by using the Bernoulli process as

• Stochastic process