Random variable
In the Stochastic is a random variable or random variable is (rare stochastic variable or stochastic variable ) is a variable whose value depends on luck. A random variable can be formally described as a function that maps the results of a random experiment values (called realizations ).
A random variable formalized so that a random experiment, which generate random numbers results. Based on the random results, especially arithmetic operations and size comparisons are possible. Examples of random variables are the eyes of the sum of two dice thrown ( see below) and the amount won in a game of chance.
While previously, introduced by AN Kolmogorov concept of random variable was initially the usual German term, today the somewhat misleading term random variable has ( starting with the English random variable ) enforced.
- 4.1 discrete
- 4.2 constant
- 4.3 independently
- 4.4 identically distributed
- 4.5 independent and identically distributed
- 5.1 characteristics
- 5.2 steadily or continuously
- 5.3 measurability, distribution function and expected value
- 5.4 integrable and quasi- integrable
- 5.5 Example
- 5.6 Standardized awareness
Motivation of the formal concept
The function values of a random variable are dependent on the random variable representing. For example, be the random result of a coin toss. Then, for example, a bet can be modeled on the outcome of a coin toss using a random variable. Suppose there was a bet on the number, and if wagered properly, 1 EUR is paid, nothing else. Be the payout. Since the value depends on the coincidence is a random variable. It is the amount of litter results from the set of possible payout amounts:
You bet on two coin tosses both times on the head and refers to the combination of the outputs of coin tosses with, so can be, for example, examine the following random variables:
In the example, the amount of a specific interpretation. In the further development of probability theory, it is often convenient to consider the elements of as abstract representatives of chance to assign a concrete meaning without them, and then to record all random processes to be modeled as a random variable.
Definition
As a random variable is called a measurable function from a probability space into a measurement space.
A formal mathematical definition can give is as follows:
Example: Two-time roll of the dice
The experiment, with a fair dice to roll the dice twice, can be modeled with the following probability space:
- The amount of the 36 possible outcomes
- Is the power set of
- If you want to model two independent litters with a fair dice, so is given to all 36 results equally probable, so choose the probability measure as for.
The random variables ( diced number of the first cube ), ( diced number of the second cube ) and ( eyes sum of the first and second cube ) can be defined as the following functions:
Being selected for the Borel σ - algebra on the real numbers.
Comments
As a rule waiving the concrete specification of the related spaces; It is assumed that the context is clear, and the probability space in which the measuring space is meant for.
For a finite result set as the power set is usually chosen from. The requirement that the function used is measurable, then always fulfilled. Measurability is only really important if the result set contains uncountably many elements.
Some classes of random variables with certain probability and measurement rooms are used most often. These are partially inserted using alternative definitions that do not require knowledge of measure theory:
Real random variable
For real random variables of the image space is the set of real numbers equipped with the Borel - algebra. The general definition of random variables can be used in this case simplify to the following definition:
This means that the set of results, the realization of which is below a certain value, it must form an event.
In the example, the two-time Würfelns are, and each real random variable.
Multidimensional random variable
A multi-dimensional random variable is a measurable mapping for a dimension. It is also known as a random vector. Thus, at the same time is a vector of real individual random variables, all of which are defined in the same probability space. The distribution of is called the multivariate distributions of the components are also called marginal distributions. The multidimensional correlation between the expected value and variance of the expected return vector and covariance matrix.
In the example, the two-time die-rolling is a two-dimensional random variable.
A complex random variable
For complex random variables of the image space is the set of complex numbers provided with the " inherited " by the canonical Vektorraumisomorphie between and between borel σ - algebra. is a random variable if and only if the real part and imaginary part are each real random variable.
The distribution of random variables, existence
Closely linked with the more technical notion of a random variable is the notion of induced on the image space of probability distribution. Sometimes both terms are also used interchangeably. Formally, the distribution of a random variable is defined as the size of the probability measure, ie
Instead in the literature for the distribution of are also the spellings or used.
Thus one speaks, for example, of a normally distributed random variables, so therefore is a random variable with values in the real numbers meant, whose distribution corresponds to a normal distribution.
Properties which can be expressed solely on common distributions of random variables, also called probabilistic. For treatment of such characteristics, it is not necessary to know the actual shape of the (background) probability space, where the random variables are defined.
Often, therefore, only the distribution function is specified and allowed the probability space underlying open of a random variable. This is allowed from the viewpoint of mathematics, if there really is a probability space, which can generate a random variable with the given distribution. Such a probability space but is easy to specify a concrete distribution by, for example, as the Borel σ - algebra is chosen on the real numbers and as induced by the distribution function Lebesgue - Stieltjes measure. As a random variable can then be selected with the identity map.
If a flock is considered of random variables, it is sufficient probabilistic perspective just to specify the joint distribution of the random variable, the shape of the probability space can again be left open.
The question of the specific form of the probability space that is pushed into the background, but it is of interest whether for a family of random variables with given finite joint distributions exists a probability space on which they can be defined together. This question is for independent random variables by an existence theorem of É. Borel resolved, the says that you can rely on the probability space formed by the unit interval and Lebesgue measure in principle. Uses A possible proof that the binary digits of real numbers in [0,1] can be regarded as nested Bernoulli sequences ( similar to Hilbert's Hotel).
Mathematical attributes for random variable
Various mathematical attributes that are borrowed generally those for general functions, find random variable application. The most common are briefly explained in the following compilation:
Discrete
A random variable is called discrete if it takes only finitely many or countably infinitely many values . In the above example, the two-time die-rolling, all three random variables, and discreet. Another example of discrete random variables are random permutations.
Constant
A random variable is called a constant if it accepts only one value: for all. It is a special case of the discrete random variable.
Independent
Two real random variables are called independent if for any two intervals are and the events and stochastically independent. These are they, if:.
In the above example, and independently; the random variables and are not.
Independence of several random variable means that the probability of the random vector corresponding to the product dimension of the probability measurements of the components, ie the product dimension of. Thus, for example, three times independent cubes by the probability space with
Model; the random variable " result of th litter" is then
Constructing a probability space corresponding to any family of independent random variable with given distributions are also possible.
Identically distributed
Two or more random variables are called identically distributed (or id for identically distributed ) if their distributions are the same. Are In the example of the two-time dicing, identically distributed; the random variables and are not.
Independent and identically distributed
Often sequences of random variables are examined, which are both independent and identically distributed; This case is usually with u.i.v. or i.i.d. ( for independent and identically distributed ) abbreviated.
In the above example are the three-time dicing, and iid The sum of the first two throws and the sum of the second and third litter are indeed identically distributed but not independent. In contrast, distributed and independent, but not identical.
Mathematical attributes for real random variable
Parameters
For the characterization of random variables serve a few functions that describe the essential mathematical properties of the random variable. The most important of these functions is the distribution function, which gives information about the probability that the random variable assumes a value up to a predetermined limit, for example, the probability of rolling a maximum of four. For continuous random variables, this is supplemented by the probability density with which the probability can be calculated that the values of a random variable are within a certain interval. Furthermore, metrics such as the expected value, variance and higher moments of mathematical interest.
Continuously or continuously
The constant attribute is used for different properties.
- A real random variable is called continuous ( or absolutely continuous ), if it has a density ( their distribution is absolutely continuous with respect to Lebesgue measure.)
- A real random variable is known as continuous if it has a continuous distribution function. In particular, this means that applies to everyone.
Measurability, distribution function and expected value
If a real random variable is added to the sample space and a measurable function, then is also a random variable on the same sample space, because the link of measurable functions is measurable again. is also referred to as a transformation of the random variables under. The same method by which you can go from a probability space after, can be used to obtain the distribution of.
Is the distribution function of
The expected value of a quasi- integrable random variable of calculated according to the following:
Integrable and quasi- integrable
A random variable is called integrable if the expected value of the random variable exists and is finite. The random variable is called quasi- integrable if the expected value exists, but is potentially infinite. Each integrable random variable is thus quasi- integrable.
Example
It is a real continuous random variable and.
Then
Case distinction according to:
Standardized awareness
A random variable is called standardized if their expected value 0 and its variance is 1. The transformation of a random variable in a standardized random variable
Is called standardizing the random variable.
Others
- Time- related random variables can also be interpreted as a stochastic process
- A sequence of realizations of a random variable is also called random sequence
- A random variable generates a σ - algebra, with the Borel σ - algebra is.