Random variable

A random variable is a function definite on the whole of the possible results of a random experiment, such as it is possible to determine the probability so that it takes a given value or that it takes a value in a given interval. In the beginning, a variable was a function of profit, which represented the profit obtained at the conclusion of the result of a play. For example, let us suppose that a player launches a die and that this one gains 1€ if it brings one six and loses 10€ if it brings another result. Then it is possible to define the random variable of profit which associates 1 with the result “six” and -10 in an uninteresting result. The probability so that the random variable takes value 1 corresponds exactly to the probability so that the player gains 1€. The random variables are very much used in Theory of probability and Statistiques. In the applications, the random variables are used to model the result of a deterministic mechanism not or as the result of a deterministic Expérience not which generates a result Aléatoire.

Details

• the random variable simplest is given by the result of a throw to the play of pile or face, which is worth pile or face . Another simple example is given by the result of a jet of dice, for which the possible values are 1 , 2 , 3 , 4 , 5 , 6 (if the die is classically cubic). Such random variables are described as discrete because they take quite separate values. A contrario , the measurement of the size of an individual taken randomly in a Population resembles a positive Real number more (that is not completely true either, bus of the ergonomic questions make all the more improbable the statement of a number which it comprises of significant decimals; the attractile one is in fact a fractal). This random variable is then qualified by convention of continues . The study of the distribution of the values taken by a random variable led to the concept of Law of probability.

• In mathematics, and more precisely in Theory of probability, a random variable is a measurable Fonction definite on a space of probabilities. Corresponding image measurement is called law random variable. This type of function makes it possible to model a phenomenon Aléatoire, as for example the result of a jet of Des. an interesting property of the integral of Lebesgue makes that an event of strictly null probability is not necessarily impossible in a strict sense term (thus, consider a drawn reality randomly in interval 1; the probability that it is rational is null, whereas the rational ones constitute in this interval an infinite unit, and even everywhere dense).

Some random variables

As an introduction to the definitions concerning the random variables, it seems interesting to briefly introduce a family of variables very much used.

In addition to the variable a certain which takes value given with a probability equalizes to 1, the random variable simplest is called variable of Bernoulli. This one can take two states, which it is always possible to code 1 and 0, with the probabilities p and 1 - p . A simple interpretation relates to a play of die in which one would gain one euro by drawing the six ( p = 1/6). On a sequence of parts, the average of the profits tends towards p when the number of parts tends towards the infinite one.

If it is considered that a part is consisted N pullings instead of only one, the total of the profits is a realization of a binomial variable which can take all the whole values of 0 with N . This variable has as an average produces it Np . One obtains a less futile example by considering the score of a candidate in a electoral survey.

If N is rather large and p not too small, one can find an approximation suitable by using the variable Gauss. In the surveys that makes it possible to associate a confidence interval with the gross profit. Thus, there are 95 chances out of 100 so that an investigation carrying into: 1000 people gives a correct result except for ± 3%.

Always with large N , the approximation of Poisson is preferable if p is rather small so that the average Np is not too large, about a few units. In a survey it would be the law applicable to the “small” candidates. It is especially the law used in queueing problems.

The sum of the variable squares of ν of Gauss independent is a variable of χ² to ν degrees of freedom (the exponential variable is a particular case). The test of the χ² is used to appreciate the value of the adequacy of a law of probability on an empirical distribution.

If one divides a variable of Gauss by a variable of χ (square root of the preceding one), one obtains a variable of Student. The ratio of two independent variables of χ² defines a Variable of Snedecor. These two laws are used in the analysis of presumedly Gaussian populations.

Basic concepts

Function of distribution

It would be possible to introduce this notion from any of the variables previously considered but it appears clearer to study the case of the die under a different angle. Indeed, it defines a random variable X which takes with the same probability of appearance (1/6) of the values as a whole {1,2,3,4,5,6}. One can then associate with any actual value X the probability of obtaining a pulling lower or equal to X, which defines a curve in staircase whose steps have a height equalizes to 1/6.

Formally, that led to a Function of distribution

$F\_X\; (X)\; =\; P\; (X\; \backslash \; Leq\; X)\; \backslash ,$

In this one, capital letter X represents the random variable, together of numerical values, and tiny X represents the variable of state, variable in the usual sense of the term.

If the events are not equiprobable any more, that does nothing but deform the curve. To introduce a new concept, one can start by replacing it from a caster with six numbers (what leads to a rigorously identical problem). Then, nothing fundamental is changed if one replaces the six integers by the reference marks of the centers of arcs of 60 degrees. From there it is possible to increase the number of sectors by reducing their size: the levels will become increasingly small until being indistinguishable on a drawing. The passage in extreme cases replaces the discrete variable by a continuous variable which takes all the actual values in the interval] 0,360]: it is a uniform variable.

A Fonction of distribution is increasing (in the broad sense) on the interval] - ∞, +∞ and continues on the right in any point; it tends towards 0 in - ∞ and 1 in +∞. Reciprocally, any function checking the properties (characteristic) preceding can be regarded as the function of distribution of a random variable.

The interest of the Fonction of distribution lies in the fact that it is valid as well for the continuous variables definite on a continuous whole as for the discrete variables definite on a countable unit (in the majority of the practical cases it is reduced to a whole of equidistant values which one can bring back to a whole of entireties). The progressive replacement of curves in staircases by continuous curves makes it possible to see intuitively how a continuous variable can provide an approximation often easier to handle than the original discrete variable.

Unfortunately these benefits of the Function of distribution, interesting for the visualization of the phenomena, are lost as soon as one wants to look further into the problems. In this case, it is generally more convenient to use the concepts described in the following paragraphs.

Function of probability of a discrete variable

The law of a discrete variable is quite simply given by the whole of the probabilities of its values (Function of Mass). If it is supposed that it takes whole values (of unspecified sign), that is written:

$P\_X\; (I)\; =\; P\; (X\; =\; I)\; \backslash \; (I\; \backslash \; in\; \backslash \; mathbb\; \{Z\})$

One rebuilds the function of distribution (of which the values are then called cumulated probabilities ) by the relation:

if , then $F\_X\; (X)\; =\; \backslash \; sum\_\; \{k=-\; \backslash \; infty\}\; ^n\; P\_X\; (K)$.

Density of probability of a continuous variable

A continuous variable in general has a function of distribution continuous in any point and derivable per pieces. It is then convenient to derive it to obtain the density of probability, checking:

$p\_X\; (X)\; =\; \{dF\_X\; \backslash \; over\; dx\}$

who is defined and with values positive (or null) on $]\; -\; \backslash \; infty,\; \backslash ,\; +\; \backslash \; infty\; such\; as$ \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; p\_X\; (X)\; \backslash \; dx\; =\; 1$.$

One rebuilds the function of distribution by the relation:

$F\_X\; (X)\; =\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{X\}\; p\_X\; (\backslash \; xi)\; \backslash \; D\; \backslash \; xi$

In the general reasoning, it is often convenient to write these formulas in differential form:

If one carries out a change of variable according to the formula $Y\; =\; F\; (X)\; \backslash ,$, the new density of probability is calculated by:

$p\_Y\; (there)\; Dy\; =\; p\_X\; (X)\; dx\; \backslash ,$

Expectation

Definition

The mathematical Espérance of a random variable is defined as the average values taken by this variable, balanced by their probabilities.

In the case of a discrete variable that one supposes to take the whole values, it is defined simply by

$E\; =\; \backslash \; sum\_\; \{k=-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; K\; \backslash \; P\_X\; (K)$ (subject to existence, i.e. here of sommability).

For a continuous variable, the differential formula given previously is integrated in

$E\; =\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; X\; \backslash \; p\_X\; (X)\; \backslash \; dx$ (subject to existence, i.e. here of integrability).

Generalization: hope of a function of a random variable

X being a random variable, a presumedly regular function F defines a new random variable F (X) whose hope, when it exists, is written by replacing K by F (K) or X by F (X) in the preceding formulas (theorem of transfer).

Discrete random variable: $E\; =\; \backslash \; sum\_\; \{k=-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (K)\; \backslash \; P\_X\; (K)$

Continuous random variable: $E\; =\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (X)\; \backslash \; p\_X\; (X)\; \backslash \; dx$

In particular, it is interesting to consider the random variable with complex values $e^\; \{I\; \backslash \; theta\; X\}\; \backslash ,$ (where $\backslash \; theta\; \backslash \; in\; \backslash \; mathbb\; \{R\}$) whose expectation is value in $\backslash \; theta$ transform of opposite Fourier of the density of probability:

$\backslash \; phi\_X\; (\backslash \; theta)\; =\; E\; \backslash \; left\; \backslash \; theta\; X\}\; \backslash \; right\; \backslash ,$

The exponential one develops in series: $\backslash \; phi\_X\; (\backslash \; theta)\; =\; E\; \backslash \; left\; \{(I\; \backslash \; theta\; X)\; ^k\; \backslash \; over\; \{K!\}\}\; \backslash \; right$

or, if the density of probability is a sufficiently regular function:

$\backslash \; phi\_X\; (\backslash \; theta)\; =\; \backslash \; sum\_\; \{k=0\}\; ^\; \backslash \; infty\; \{(I\; \backslash \; theta)\; ^k\; \backslash \; over\; \{K!\}\}\; E\; \backslash \; left$

Moments of a random variable

Definition

One defines the moment of order K of a random variable X , which one notes $m\_k$, like:

$m\_k\; \backslash \; equiv\; E\; \backslash ,$.

The moments are useful to characterize a random variable:

• moment of order one of the variable: $\backslash \; driven\; \backslash \; equiv\; m\_1=\; E\; \backslash ,$ corresponds to the hope

• moment of order two of the variable centered : $\backslash \; sigma^2\; \backslash \; equiv\; m\_2\; =\; E\; \backslash \; left\; \backslash ,$ is called variance.

• moment of order three of the center-reduced variable : $m\_3\; =E\; \backslash \; left\; \backslash \; left\; (\backslash \; frac\; \{X\; \backslash \; driven\}\; \{\backslash \; sigma\}\; \backslash \; right)\; ^3\; \backslash \; right\; \backslash ,$ is called coefficient of asymmetry.

• moment of order four of the center-reduced variable : $m\_4\; =E\; \backslash \; left\; \backslash ,$ is called Kurtosis.

Interpretation of the moments

• the hope is an indicating of central tendency .

• the variance is an indicating of dispersion . It is often noted $\backslash \; sigma^2$. The root square of the variance, called standard deviation, and noted $\backslash \; sigma$, provides a homogeneous size to the basic size and the average

• the coefficient of asymmetry is an indicating of asymmetry .

• the kurtosis is an indicating of flatness of the extremes of the distributions .

Characterization of a random variable by its moments

Under certain conditions of continuity on a closed and limited segment, a sufficiently regular random variable can be characterized, and in an equivalent way, by its function of distribution, its density of probability (or its function of probability), and by its moments $m\_k\; (X)\; \backslash ,$. See Moment (mathematics).

Formulas of recursive determination of the moments

By defining

• moments compared to the origin (ordinary moments or raw moments in English):

$m\_k\; (X)\; \backslash \; equiv\; E\; \backslash \; left\; \backslash ,$.

• the central moments, generally noted $\backslash \; mu\_k\; (X)\; \backslash ,$ and which are defined as follows:

$\backslash \; mu\_k\; (X)\; \backslash \; equiv\; E)\; ^k]\; \backslash ,$

There exist formulas (which resemble that of the binomial) making it possible to calculate one central moment of order K as from the ordinary moments of a nature lower or equal to K, and reciprocally; here some examples (until order 4):

$\backslash \; mu\_2\; =\; m\_2\; -\; m^2\_1\; \backslash ,$

$\backslash \; mu\_3\; =\; m\_3\; -3\; \backslash ,\; m\_1\; \backslash ,\; m\_2\; +\; 2m^3\_1\; \backslash ,$

$\backslash \; mu\_4\; =\; m\_4\; -4\; \backslash ,\; m\_1\; \backslash ,\; m\_3\; +\; 6\; \backslash ,\; m^2\_1\; \backslash ,\; m\_2\; -\; 3m^4\_1\; \backslash ,\; \backslash !$

and:

$m\_2\; =\; \backslash \; mu\_2\; +\; m^2\_1\; \backslash ,$

$m\_3\; =\; \backslash \; mu\_3\; +\; 3\; \backslash ,\; m\_1\; \backslash \; mu\_2\; +\; m^3\_1\; \backslash ,$

$m\_4\; =\; \backslash \; mu\_4\; +\; 4\; \backslash ,\; m\_1\; \backslash \; mu\_3\; +\; 6\; \backslash ,\; m^2\_1\; \backslash \; mu\_2\; +\; m^4\_1\; \backslash ,$

Median and quantiles

One calls median of a random variable X, a reality m such as

$P\; (X\; \backslash \; Leq\; m)\; \backslash \; geq\; 1/2\; \backslash \; Leq\; P\; (X\; \backslash \; geq\; m)$

In the case of a discrete random variable , this definition is not very interesting because it allows the existence of several medians

if X is the number appearing on the higher face of a perfectly balanced die with 6 faces, for any reality m strictly ranging between 3 and 4, one a:

$P\; (X\; \backslash \; Leq\; m)\; =\; P\; (X\; \backslash \; geq\; m)\; =\; 1/2$

or the existence of a median which does not give a probability of 0,5

If X is the sum obtained while launching two perfectly balanced dice with 6 faces. X has only one median 7 but $P\; (X\; \backslash \; Leq\; 7)\; =\; 21/36$

Danslecasde a variable continues , if the function of distribution is strictly increasing, the definition is equivalent to the following one:

the median of X is single reality m such as $F\_X\; (m)\; =0,5\; \backslash ,$

The fact that the function of distribution is continuous, and supposed strictly increasing, with values in] 0  ;   1 ensures the existence and the unicity of the median.

If the median has like value m=0.5, it is possible however to be interested in others valeursde m (which one names the Quantiles:

• Quartile: m = 0.25,0.75
• Decile: m = 0.1,0.2,0.3…
• Centile: m = 0.01,0.02…

Data processing

One often uses generating pseudo random for to simulate the chance. There exist also means of exploiting the indetermination of physical phenomena, for example by analyzing the variations of a film of Lampe to lava, or by analyzing the thermal Bruit.

See too

• Convergence of random variables
• Law of probability to several variables
• Generating of pseudo-random numbers
• Generating of random random numbers
• Law of probability
• Random variable elementary
• Contained Random

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