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Geometric distribution

Lecture



Geometric distribution
Probability function
Geometric distribution
Distribution function
Geometric distribution
Designation Geometric distribution
Options Geometric distribution —Number of “failures” before the first “success”
Geometric distribution - the probability of "success"
Geometric distribution - probability of “failure”
Geometric distribution —Number of the first “success”
Geometric distribution - the probability of "success"
Geometric distribution - probability of “failure”
Carrier Geometric distribution Geometric distribution
Probability function Geometric distribution Geometric distribution
Distribution function Geometric distribution Geometric distribution
Expected value Geometric distribution Geometric distribution
Median N / A N / A
Fashion Geometric distribution Geometric distribution
Dispersion Geometric distribution Geometric distribution
Asymmetry coefficient Geometric distribution Geometric distribution
Coefficient of kurtosis Geometric distribution Geometric distribution
Informational entropy Geometric distribution Geometric distribution
Generating function of moments Geometric distribution Geometric distribution
Characteristic function Geometric distribution Geometric distribution

Geometric distribution in probability theory is the distribution of a discrete random value equal to the number of trials of a random experiment prior to the observation of the first “success”.

Definition

Let be Geometric distribution - an infinite sequence of independent random variables with the distribution of Bernoulli, that is,

Geometric distribution

Construct a random variable Geometric distribution - the number of “failures” to the first “success”. Random distribution Geometric distribution called geometric with probability of "success" Geometric distribution what is denoted as follows: Geometric distribution .

Probability function of a random variable Geometric distribution has the form:

Geometric distribution

Note

  • It is sometimes assumed by definition that Geometric distribution - the number of the first "success". Then the probability function takes the form Geometric distribution . The table on the right shows the formulas for both options.
  • The probability function is a geometric progression, hence the name of the distribution.

Moments

The generating moment function of the geometric distribution has the form:

Geometric distribution ,

from where

Geometric distribution ,

Geometric distribution .

Geometric Distribution Properties

  • Of all discrete distributions with carrier Geometric distribution and fixed average Geometric distribution geometric distribution Geometric distribution is one of the distributions with maximum informational entropy.
  • If a Geometric distribution independent and Geometric distribution then

Geometric distribution .

  • The geometric distribution is infinitely divisible.

Lack of memory [edit]

If a Geometric distribution then Geometric distribution , that is, the number of past “failures” does not affect the number of future “failures”.

Geometric distribution is the only discrete distribution with the lack of memory property.

Relationship with other distributions

  • Geometric distribution is a special case of a negative binomial distribution: Geometric distribution .
  • If a Geometric distribution independent and Geometric distribution then

Geometric distribution .

Example

Let the dice roll before the first six falls out. Then the probability that we will need no more than three shots is equal to

Geometric distribution

Geometric distribution .

The expected number of shots is

Geometric distribution .

Geometric distribution. Examples

The geometric distribution law takes place in such sciences as microbiology, genetics, physics. In practice, the experiment or experience is carried out before the first occurrence of a successful event A. The number of attempts made will be an integer random variable 1,2, .... The probability of occurrence of event A in each experiment does not depend on the previous ones and is p, q = 1-p. The probabilities of possible values ​​of a random variable X are determined by the dependence


Geometric distribution


There is in all previous experiments except for the k-th the experiment gave a bad result and only in the k-th was it successful. This probability formula is called the geometric distribution law, since its right-hand side coincides with the expression of a common element of a geometric progression.

In tabular form, the geometric law of distribution is


Geometric distribution

When checking the normalization condition, we use the formula of the sum of an infinite geometric progression

Geometric distribution

The probabilistic generating function is expressed by the formula

Geometric distribution

Insofar as Geometric distribution then the generating function can be summed

Geometric distribution

Geometric distribution

Numerical characteristics for the geometric law of probability distribution is determined by the formulas:

1. Mathematical expectation

Geometric distribution

Geometric distribution
Geometric distribution

2. Dispersion and standard deviation by the formulas

Geometric distribution

Geometric distribution

Geometric distribution

Geometric distribution
Geometric distribution

3. The coefficient of asymmetry and kurtosis for the geometric distribution is determined by the formula

Geometric distribution

Geometric distribution

Among discrete random variables, only the geometric law is given the property of the absence of aftereffect. This means that the probability of occurrence of a random event in the k-th experiment does not depend on how many of them appeared before the k-th, and is always equal to p.

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Example 1. A die is thrown before the first appearance of the number 1. Determine all the numerical characteristics of M (X), D (X), S (X), A (X), E (X) for a random variable X of the number of flips to be performed.

Decision. By the condition of the problem, the random variable X is integral with the geometric law of probability distribution. The probability of a successful flip is a constant and equal to one divided by the number of faces of the cube.

Geometric distribution

Having p, q necessary numerical characteristics of X we find the above formulas

Geometric distribution

Geometric distribution

Geometric distribution

Geometric distribution

Geometric distribution

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Example 2. Amateur hunter shoots a gun at a fixed target. The probability of hitting the target with one shot is constant and equal to 0.65. Shooting at the target is before the first hit.

Determine the numerical characteristics M (X), D (X), S (X), A (X), E (X) of the number of cartridges spent by the hunter.

Decision. The random variable X obeys the geometric distribution law, therefore the probability of hitting in each attempt is constant and is p = 0.65; q = 1-p = 0.35.

By the formulas we calculate the expectation

Geometric distribution

dispersion

Geometric distribution

standard deviation

Geometric distribution

asymmetry

Geometric distribution

excess

Geometric distribution

The calculation of numerical characteristics for the geometric distribution law is not so complicated, so use the formulas given in such problems and get only the correct results.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis