Marginal distribution. Excerpt characterizing Marginal distribution

Intuitively, marginal probability X calculated by studying full probability X for a certain value Y, and then averaging this conditional probability over the distribution of all values Y.

This follows from the definition mathematical expectation, that is, in general case:

Real example

Suppose you want to calculate the probability that a pedestrian will be hit by a car while crossing the road at a crosswalk, ignoring traffic lights. Let H be a discrete random variable taking one of the values ​​(Knocked down, Not knocked down). Let L be a discrete random variable taking the values ​​of the traffic light color (Red, Yellow, Green).

In reality, H should depend on L. That is, P(H = Shot down) and P(H = Not shot down) takes different meanings, depending on what color the traffic light L is on: red, yellow, or green. For example, a person is more likely to be hit by a car when trying to cross the road when the car's light is green, and less likely when the car's light is red. In other words, for each possible pair of values ​​of H and L, one must consider the joint probability distribution of H and L to find the probability that the pair of events will occur together if the pedestrian ignores the color of the traffic light.

However, when trying to calculate the marginal probability P(H=Knocked down), in which we require that the probability H=Knocked down in a situation where we do not know the value of L and when the pedestrian ignores the color of the traffic light. In general, a pedestrian can be hit when the light is red OR yellow OR green. Thus, in this case the marginal probability can be found by summing P(H,L) for all possible values L, where each L value is weighted by its probability of occurrence.

Below is a table of the conditional probability that a pedestrian will be hit, depending on the traffic light signal. (Note that the columns in this table must add up to 1 since the probability of being hit or not being hit is 1 regardless of the state of the traffic light.)

To find the joint probability distribution, we need more data. Let P(L=red) = 0.2, P(L=yellow) = 0.1, and P(L=green) = 0.7. By multiplying each column in the conditional probability by the probability that a pedestrian is hit by a given traffic signal, we find the joint distribution of H and L, which is written in row 3. (Note that summing the values ​​of row 3 gives one).

Marginal probability P(H=Knocked down) is the sum of the values ​​in the row H=Knocked down of the table joint distribution, since it is the probability of being shot down when the signal is red or yellow or green. Similarly, the marginal probability that P(H=Not shot down) is the sum of the values ​​of the line H=Not shot down. In this example, the probability of being hit by an inattentive pedestrian is 0.572.

Multivariate distributions

The formula for multivariate distribution is similar to those given above, in which the symbols X and/or Y are interpreted as vectors. In particular, each summation or integration will take place over all variables except those contained in X.

See also

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Links

Literature

• Everitt, B.S. The Cambridge Dictionary of Statistics. - Cambridge University Press, 2002. - ISBN 0-521-81099-X.
• Trumpler, Robert J. and Harold F. Weaver. Statistical Astronomy. - Dover Publications, 1962.

Excerpt characterizing Marginal distribution

Consider a positive definite quadratic form. Let the size vector be split into two parts after the th element, so that , and let the size matrix also be split into parts after the th row and th column, so that

.

Then, since

Can always be represented as the sum of two quadratic forms and containing, respectively, elements, where

(A7.1.1)

The determinant can be represented as

P7.1.2. Two useful integrals

Let be positive definite quadratic form from of the elements, so , where , and let and be positive real numbers. Then it can be shown that

, (A7.1.3)

Where -multiple integral is taken over the entire space of possible and

, (A7.1.4)

where the function is called the -distribution with and degrees of freedom and is defined as

,

and, denoting , we obtain from (A7.1.4)

, (A7.1.6)

where the function is called the -distribution with degrees of freedom and is defined as

. (A7.1.7)

P7.1.3. Normal distribution

A random variable is said to be normally distributed with mean and standard deviation or to have a distribution if its probability density is

Hence the normalized quantity has a distribution. Table E at the end of the book gives the ordinates and values ​​for which for a given.

Multivariate normal distribution. A vector random variable is said to be has a joint -dimensional normal distribution if the probability density is

Probability density isosurfaces are ellipsoids defined by the equations .

Rice. P7.1. Isolines of two-dimensional normal distribution(1); The marginal distribution (2) and the distribution condition under (3) are also shown there.

As an illustration in Fig. A7.1 shows elliptical isolines for a two-dimensional normal distribution.

At a point, the multivariate distribution has a maximum probability density

Distribution as the probability of not falling into the area limited by the isosurface of the multivariate normal distribution. For a -dimensional normal distribution (A7.1.9), the probability of not falling into the area bounded by the surface given by the equation

,

equal to - integral with degrees of freedom

,

where the distribution density is determined by formula (A7.1.7). In table F at the end of the book there are values ​​for which, for a given .

Marginal and conditional distributions for the multivariate normal distribution. Let us assume that the vector of random variables is divided into two parts after the th element, so that the covariance matrix has the form

.

Then, using (A7.1.1) and (A7.1.2), we can write the multivariate normal distribution for quantities as the marginal distribution multiplied by the conditional distribution for a given i.e.

(P7.1.10)

(P7.1.11)

and defines a regression hyperplane in -dimensional space that traces the mean point of the elements as the elements vary. The size of the regression coefficient matrix is ​​determined by the formula.

Both the marginal and conditional distributions for the multivariate normal law are themselves multivariate normal distributions. It can be seen that for a multivariate normal distribution the conditional distribution is preserved up to a shift for any .

One-dimensional marginal densities. In particular, marginal density for one element equal to - a one-dimensional normal density with mean equal to the th element and variance equal to the th diagonal element.

Bivariate normal distribution. As an example in Fig. A7.1 shows the marginal and conditional distributions for the bivariate normal distribution. In this case, the marginal distribution is , and the conditional distribution for this is equal to

,

Where - correlation coefficient between and .

P7.1.4. Student distribution

A random variable is said to have a normalized Student t-distribution with mean , normalizing parameter, and degrees of freedom if

From here standard deviation has a distribution. In table G at the end of the book there are values ​​for which, for a given .

Transition to normal distribution. For larger pieces

tends to unity, while the rightmost factor in (A7.1.12) tends to . Therefore, if we accept for large ones, then the -distribution tends to the normal distribution (A7.1.8).

Multivariate distribution. Let it be a vector size and is a positive definite matrix of size . A vector random variable is said to have a normalized -distribution with a vector of means, a normalizing matrix, and degrees of freedom if

Isosurfaces of the probability density of a multivariate -distribution are ellipsoids defined by the equation

.

Transition to multivariate normal distribution. For larger pieces

tends to unity; the rightmost bracket in (A7.1.13) tends to . Hence, if we accept for large values, the multivariate -distribution tends to the multivariate normal distribution (A7.1.9).

F-distribution as the probability of not falling into the area limited by the isosurface of the multidimensional distribution. Using (A7.1.4), we can express the probability of not falling into the region limited by the isosurface of the -dimensional distribution, given by the equation

,

as -integral with and degrees of freedom

,

where the density function for is defined by formula (A7.1.5). For big ones, where. Hence, as one would expect, the probability of not falling into the area limited by a given isosurface -distribution, for large ones, is equal to the similar probability for a multidimensional normal distribution, to which a multidimensional distribution tends.. -distributions having degrees of freedom, while conditional distributions - These are distributions with degrees of freedom. Next, the normalizing factor for conditional distribution, for example, depends on . This is a clear difference from the conditional distribution for the normal case, where the variance does not depend on .

$p\_(X)(x)\; =\; \backslash int\_y\; p\_(X|Y)(x|y)\; \backslash ,\; p\_Y(y)\; \backslash ,\; \backslash operatorname(d)\backslash !y\; =\; \backslash mathbb(E)\_(Y)$

Intuitively, marginal probability X calculated by examining the total probability X for a certain value Y, and then averaging this conditional probability over the distribution of all values Y.

This follows from the definition of mathematical expectation, that is, in the general case:

$\backslash mathbb(E)\_Y\; =\; \backslash int\_y\; f(y)\; p\_Y(y)\; \backslash ,\; \backslash operatorname(d)\backslash !y$

Real example

Suppose you want to calculate the probability that a pedestrian will be hit by a car while crossing the road at a crosswalk, ignoring traffic lights. Let H be a discrete random variable taking one of the values ​​(Knocked down, Not knocked down). Let L be a discrete random variable taking the values ​​of the traffic light color (Red, Yellow, Green).

In reality, H should depend on L. That is, P(H = Knocked Down) and P(H = Not Knocked Down) take on different values, depending on what color the traffic light L is on: red, yellow, or green. For example, a person is more likely to be hit by a car when trying to cross the road when the car's light is green, and less likely when the car's light is red. In other words, for each possible pair of values ​​of H and L, one must consider the joint probability distribution of H and L to find the probability that the pair of events will occur together if the pedestrian ignores the color of the traffic light.

However, when trying to calculate the marginal probability P(H=Knocked down), in which we require that the probability H=Knocked down in a situation where we do not know the value of L and when the pedestrian ignores the color of the traffic light. In general, a pedestrian can be hit when the light is red OR yellow OR green. Thus, in this case, the marginal probability can be found by summing P(H,L) for all possible values ​​of L, where each value of L is weighted by its probability of occurrence.

Below is a table of the conditional probability that a pedestrian will be hit, depending on the traffic light signal. (Note that the columns in this table must add up to 1 since the probability of being hit or not being hit is 1 regardless of the state of the traffic light.)

To find the joint probability distribution, we need more data. Let P(L=red) = 0.2, P(L=yellow) = 0.1, and P(L=green) = 0.7. By multiplying each column in the conditional probability by the probability that a pedestrian is hit by a given traffic signal, we find the joint distribution of H and L, which is written in row 3. (Note that summing the values ​​of row 3 gives one).

The marginal probability P(H=Knocked Down) is the sum of the values ​​in the H=Knocked row of the joint distribution table, since it is the probability of being shot down when the signal is red or yellow or green. Similarly, the marginal probability that P(H=Not shot down) is the sum of the values ​​of the line H=Not shot down. In this example, the probability of being hit by an inattentive pedestrian is 0.572.

Multivariate distributions

The formula for multivariate distribution is similar to those given above, in which the symbols X and/or Y are interpreted as vectors. In particular, each summation or integration will take place over all variables except those contained in X.

See also

Write a review about the article "Marginal distribution"

Links

Literature

• Everitt, B.S. The Cambridge Dictionary of Statistics. - Cambridge University Press, 2002. - ISBN 0-521-81099-X.
• Trumpler, Robert J. and Harold F. Weaver. Statistical Astronomy. - Dover Publications, 1962.

	n Probability distributions
	One-dimensional	Multidimensional
Discrete:	Bernoulli | Binomial | Geometric | Hypergeometric | Logarithmic | Negative binomial | Poisson | Discrete uniform	Multinomial
Absolutely continuous:	Beta | Weibull | Gamma | Hyperexponential | Gompertz distribution | Kolmogorov | Cauchy | Laplace | Lognormal | Normal (Gaussian) | Logistics | Nakagami | Pareto | Pearson | Semicircular | Continuous uniform | Rice | Rayleigh | Student's test | Tracy - Vidoma | Fisher | Chi-square | Exponential | Variance-gamma	Multivariate normal | Copula

Excerpt characterizing Marginal distribution

At half past five Napoleon rode on horseback to the village of Shevardin.
It was beginning to get light, the sky cleared, only one cloud lay in the east. Abandoned bonfires burned out in low light morning.
A thick, lonely cannon shot rang out to the right, rushed past and froze in the midst of general silence. Several minutes passed. A second, third shot rang out, the air began to vibrate; the fourth and fifth sounded close and solemnly somewhere to the right.
The first shots had not yet sounded when others were heard, again and again, merging and interrupting one another.
Napoleon rode up with his retinue to the Shevardinsky redoubt and dismounted from his horse. The game has begun.

Returning from Prince Andrei to Gorki, Pierre, having ordered the horseman to prepare the horses and wake him up early in the morning, immediately fell asleep behind the partition, in the corner that Boris had given him.
When Pierre fully woke up the next morning, there was no one in the hut. Glass rattled in the small windows. The bereitor stood pushing him aside.
“Your Excellency, your Excellency, your Excellency...” the bereitor said stubbornly, without looking at Pierre and, apparently, having lost hope of waking him up, swinging him by the shoulder.
- What? Has it started? Is it time? - Pierre spoke, waking up.
“If you please hear the firing,” said the bereitor, a retired soldier, “all the gentlemen have already left, the most illustrious ones themselves have passed a long time ago.”
Pierre quickly got dressed and ran out onto the porch. It was clear, fresh, dewy and cheerful outside. The sun, having just broken out from behind the cloud that was obscuring it, splashed half-broken rays through the roofs of the opposite street, onto the dew-covered dust of the road, onto the walls of the houses, onto the windows of the fence and onto Pierre’s horses standing at the hut. The roar of the guns could be heard more clearly in the yard. An adjutant with a Cossack trotted down the street.
- It's time, Count, it's time! - shouted the adjutant.
Having ordered his horse to be led, Pierre walked down the street to the mound from which he had looked at the battlefield yesterday. On this mound there was a crowd of military men, and the French conversation of the staff could be heard, and the gray head of Kutuzov could be seen with his white cap with a red band and the gray back of his head, sunk into his shoulders. Kutuzov looked through the pipe ahead along the main road.
Entering the entrance steps to the mound, Pierre looked ahead of him and froze in admiration at the beauty of the spectacle. It was the same panorama that he had admired yesterday from this mound; but now this entire area was covered with troops and the smoke of shots, and slanting rays bright sun, rising from behind, to the left of Pierre, they threw at her in the clear morning air, piercing with golden and pink tint light and dark, long shadows. The distant forests, completing the panorama, as if carved from some precious yellow-green stone, were visible with their curved line of peaks on the horizon, and between them, behind Valuev, a large Smolensk road, all covered with troops. Golden fields and copses glittered closer. Troops were visible everywhere - in front, right and left. It was all lively, majestic and unexpected; but what struck Pierre most of all was the view of the battlefield itself, Borodino and the ravine above Kolocheya on both sides of it.
Above Kolocha, in Borodino and on both sides of it, especially to the left, where in the swampy banks Voina flows into Kolocha, there was that fog that melts, blurs and shines through when the bright sun comes out and magically colors and outlines everything visible through it. This fog was joined by the smoke of shots, and through this fog and smoke the lightning of the morning light flashed everywhere - now on the water, now on the dew, now on the bayonets of the troops crowded along the banks and in Borodino. Through this fog I could see white church, in some places the roofs of Borodin's huts, in some places solid masses soldiers, here and there green boxes, guns. And it all moved, or seemed to move, because fog and smoke stretched throughout this entire space. Both in this area of ​​the lowlands near Borodino, covered with fog, and outside it, above and especially to the left along the entire line, through forests, across fields, in the lowlands, on the tops of elevations, cannons, sometimes lonely, were constantly born by themselves, out of nothing, sometimes huddled, sometimes rare, sometimes frequent clouds of smoke, which, swelling, growing, swirling, merging, were visible throughout this space.

private distribution - distribution of a random variable or set random variables, considered as a component or set of components of a certain random vector (see. Multivariate distribution)With given distribution. Otherwise, M. r. is the projection of the random vector distribution X=(X 1, . . ., X p).on any axis x 1 or a subspace defined by variables and completely determined by the distribution of that vector. For example, if F( x 1, x 2) - distribution function X=(X 1 , X 2) then the distribution function X 1 equal if bivariate distribution absolutely continuous and p( x 1, x 2). - its density, then the density of M. r. X 1 equal to

M. R. is calculated similarly. for any component or set of vector components X=(X 1, ..., X p).for any p. If the distribution of X is normal, then all M. r. are also normal. In the case when the values X 1, ..., X p mutually independent, according to M. r. component X 1, ..., X p vector X is uniquely determined by its distribution:

M. r. is determined similarly. with respect to a probability distribution defined on the product of spaces more general than the number line.

Lit.: Loev M., Theory of Probability, trans. from English, no., 1962; K r a m e r G., Mathematical methods statistics, trans. from English, , M., 1975. A. V. Prokhorov.

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• - 1. Shares of total income attributable to various strata of society. Functional is the distribution of income in accordance with the services of labor, land and capital...

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• - : An electoral district where party support is distributed fairly evenly with a slight advantage in favor of the party whose representatives occupy official positions...

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• - χ2 - distribution, given by the function density Kn=( , x > 0, 0 , x ≤ 0, where Γ is the gamma function, the parameter n is called the number of degrees of freedom. If X1, X2, ...

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• - 1. Distribution of the organization’s net profit across accounts. Some payments may be considered expenses and deducted immediately before calculating net income...

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Distribution of a random variable or a set of random variables considered as a component or set of components of a certain random vector (see Multivariate distribution)With given distribution. Otherwise, M. r. is the projection of the random vector distribution X=(X 1, . . ., X p).on any axis x 1 or a subspace defined by variables and completely determined by the distribution of that vector. For example, if F( x 1, x 2) - distribution function X=(X 1 , X 2) then the distribution function X 1 is equal if the two-dimensional distribution is absolutely continuous and p( x 1, x 2). - its density, then the density of M. r. X 1 equal to

M. R. is calculated similarly. for any component or set of vector components X=(X 1, ..., X p).for any p. If the distribution of X is normal, then all M. r. are also normal. In the case when the values X 1, ..., X p mutually independent, according to M. r. component X 1, ..., X p vector X is uniquely determined by its distribution:

M. r. is determined similarly. with respect to the probability distribution defined on the product of spaces more general than numerical .

Lit.: Loev M., Theory of Probability, trans. from English, no., 1962; Kramer G., Mathematical methods of statistics, trans. from English, , M., 1975. A. V. Prokhorov.

Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

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