Ah ok! (Nerd) Is at the following everything correct? - Binomial distribution
The binomial distribution is one of the most important discrete probability distributions. It describes the probable outcome of a series of similar and independent experiments, each with only two possible outcomes, the results of Bernoulli processes.
If the desired result of an experiment has the probability p, and the number of trials is n, the binomial distribution indicates the probability of a total of k successes. Under these conditions the experiment is a Bernoulli experiment. The formula is the following: \begin{equation*}P=\binom{n}{k}\cdot p^k\cdot (1-p)^{n-k}\end{equation*}
The binomial distribution is suitable for describing random variables of the following kind:
- The determination of the number of a particular property in a sample from a set of elements, if the sequence does not play a role when the sample is taken from the total quantity, and the extracted elements are returned (draws with replacement).
- The determination of the total number of defective components produced under identical conditions.
- Estimation of the random number of identical components which fail in a time interval when used under the same boundary conditions.
- Hypergeometric distribution
The general situation is the following:
In an urn there are $ N $ objects, $ K $ objects have a certain property, and the remaining objects do not have this property. If one takes a random sample of $ n $ objects from the urn without replacement, the individual extractions are not independent. The random variable $ Y $ specifies the number $ k $ of the objects in the sample that have the property. The probability of $Y$ is equal to:
\begin{equation*}H=\frac{\binom{K}{k}\cdot \binom{N-K}{n-k}}{\binom{N}{n}}\end{equation*}
We use the hypergeometric distribution at draws without replacement but the binomial distribution at draws with replacement.
- Geometric distribution
The geometric distribution is a kind of discrete distributions. The geometric distribution refers to the number of executions up to the first so-called "success".
At the geometric distribution we consider also the Bernoulli-Experiment.
If the success probability $ p $ has been given, the probability function
\begin{equation*}P(X=n)=p\cdot (1-p)^{n-1}\end{equation*}
can be used to determine the exact probability of exactly $ n $ attempts to reach the first success / hit.
- Poisson distribution
The Poisson distribution is one of the discrete probability distributions.
The Poisson distribution is a typical distribution for the number of phenomena occurring within a unit.
It is often used to describe temporal events. Given is a random event, which takes place on an average at a time interval $ t_1 $, as well as a second period $ t_2 $, to which this event is to be referred.
The Poisson distribution $ P_{\lambda} (n) $ with $ \lambda = \frac{t_2}{t_1} $ indicates the probability that exactly $ n $ events occur in the period $ t_2 $.
The distribution function denoted by $ P_{\lambda} $ is determined by the event rate (mean occurrence frequency of an event) parameter $ \lambda $, which is the expectation value and the variance of the distribution. It assigns the probabilities to the natural numbers $ k = 0,1,2, \ldots $ as follows:
\begin{equation*}P_{\lambda}(X=k)=\frac{\lambda^k}{k!}e^{-\lambda}\end{equation*}
Typical questions which can be answered with the help of the Poisson distribution are e.g. the
- number of events within a particular time unit (e.g., number of incoming telephone calls in a call center within an hour or number of customers in a supermarket within one hour), or
- number of objects on a given area (e.g., number of molehill on one hectare), or in a certain volume (e.g., number of bacteria in one liter of liquid).
The prerequisite for the Poisson distribution is that the events are random and independent of each other.
The Poisson distribution is also used as an approximation solution for the binomial distribution (so-called Poisson approximation), if the number of trial runs is high (eg starting at $ 100 $) and the probability of occurrence of an event is low (eg maximum $ 10% $). It is also referred to as the distribution of the rare events.
- Exponential distribution
The exponential distribution is a continuous distribution. With the aid of the exponential distribution, Life or waiting times can be modeled.
Since the exponential distribution is also used as a life distribution, it is possible to specify related variables such as survival probability, remaining life, and failure rate using the distribution function. Thus complement of the distribution function is called survival probability
\begin{equation*}P(X>x)=1-F(x)=e^{-\lambda x}\end{equation*}
The exponential distribution is used primarily in answering the question of the duration of random time intervals, e.g.
- Duration of a telephone conversation
- Duration of services, repairs, maintenance measures
- Time between two calls
- Life of atoms during radioactive decay
- Life of components, machines and devices, if aging phenomena do not have to be considered.
- Age of living creatures
Could I improve/add something? (Wondering)
