Bernoulli trial

A Bernoulli process or a Bernoulli chain (named after Jakob I Bernoulli ) is a time-discrete stochastic process that consists of a finite or countable-infinite sequence of independent experiments with Bernoulli distribution for the same parameter . This means that for each of the points in time 1, 2, 3, ... it is “rolled out” whether an event is likely to occur or not. ${\displaystyle p\in \left[0,1\right]}$ ${\displaystyle p}$

Here is an example of a possible implementation of a Bernoulli process; the symbol ♦ stands for “event occurs” (“success” for short), ◊ for “event does not occur” (“failure”), this specific sequence of events could e.g. B. at occur so that "success" is less common than "failure": ${\displaystyle p=1/3}$

◊- ♦ -◊- ♦ -◊-◊- ♦ -◊- ♦ -◊- ♦ -◊-◊-◊-◊-◊- ♦ -◊-◊-◊-◊-◊-◊-◊-…

The process can be described by a sequence of independent random variables , each of which has a constant probability of 1 ( success ) and a probability of 0 ( failure ). ${\displaystyle X_{1},X_{2},X_{3},\dotsc }$ ${\displaystyle p}$ ${\displaystyle 1-p}$

Depending on the question, you are interested in one or more of the following random variables:

• The number of successful attempts after performing a total of attempts; it follows a binomial distribution . It applies . ${\displaystyle S_{n}}$ ${\displaystyle n}$ ${\displaystyle S_{n}=X_{1}+\dotsb +X_{n}}$
• The number of attempts 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 . ${\displaystyle T_{r}}$ ${\displaystyle r}$

properties

The number of successes after attempts in a Bernoulli process is a special Markov chain : With the “time step” from to , the system is likely to go from the “state” to the state ; otherwise it remains in the state . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n+1}$ ${\displaystyle p}$ ${\displaystyle k}$ ${\displaystyle k+1}$ ${\displaystyle k}$

The random variable , which indicates how many of Bernoulli's attempts were successful, follows the binomial distribution . We derive this distribution in the following example with a cube. ${\displaystyle S_{n}}$ ${\displaystyle n}$

Examples

• When rolling the dice, the six is ​​rated as a success; so the probability of success is the complementary probability of failure . The question now is the probability of throwing exactly sixes in throws. The answer to this question can be found as follows: The probability of first throwing two sixes, then three non-sixes is . Since the order is not important, this probability has to be multiplied by the number of possibilities to distribute two (indistinguishable) six throws over five throws. The combinatorics According to this number by the binomial coefficients given "5 2"; the probability we are looking for is: ${\displaystyle p=1/6}$ ${\displaystyle 1-p=5/6}$ ${\displaystyle n=5}$ ${\displaystyle k=2}$ ${\displaystyle p^{2}(1-p)^{3}}$

${\displaystyle B(2|p,5)=\left({\begin{matrix}5\\2\end{matrix}}\right)p^{2}(1-p)^{5-2}}$.

Generalized from this, the probability of success in Bernoulli experiments is exactly one time ${\displaystyle n}$ ${\displaystyle k}$

${\displaystyle P(S_{n}=k)=B(k|p,n)=\left({\begin{matrix}n\\k\end{matrix}}\right)p^{k}(1-p)^{n-k}}$.

This function is called the binomial distribution (or binomial distribution).

• A drunk pedestrian (or a diffusing particle) is likely to move forwards on a line at every step , and backwards with a probability . For example, one is interested in the distance from the starting point. Such a model is known in physics as a one-dimensional random movement (random walk). The position of the pedestrian after steps can be represented using the Bernoulli process as ${\displaystyle p}$ ${\displaystyle 1-p}$ ${\displaystyle Y_{n}}$ ${\displaystyle n}$ ${\displaystyle (X_{k})}$

${\displaystyle Y_{n}=Y_{0}+\sum _{k=1}^{n}(2X_{k}-1)=Y_{0}+2S_{n}-n}$.

Is, for example, a realization of the Bernoulli process through the sequence

${\displaystyle (X_{n})=1,0,0,1,1,1,0,1,1,0,\ldots }$

given, then the result is for the associated random walk ${\displaystyle Y_{0}=0}$

${\displaystyle (Y_{n})=1,0,-1,0,1,2,1,2,3,2,\ldots }$.

literature

• Christian Hesse: Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , doi : 10.1007 / 978-3-663-01244-3 .