Find the probability density function of a random variable. Probability distribution density. Examples of Poisson random variables

Expected value

Dispersion The continuous random variable x, the possible values \u200b\u200bof which belong to the entire axis OH is determined by the equality:

Appointment of service. Online calculator is designed to solve problems in which either distribution density f (x), or the distribution function f (x) (see example). Usually in such tasks you want to find mathematical expectation, secondary quadratic deviation, construct graphs of functions f (x) and f (x).

Instruction. Select the type of source data: distribution density F (X) or F (X) distribution function.

The density of the distribution F (X) is specified:

The distribution function F (X) is specified:

Continuous random value is given by probability density
(The law of the relay is applied in radio engineering). Find M (x), D (x).

Casual value x called continuous if its function of the distribution f (x) \u003d p (x< x) непрерывна и имеет производную.
The function of the distribution of a continuous random variable is used to calculate the probabilities of incoming random variance at a given gap:
P (α.< X < β)=F(β) - F(α)
moreover, for a continuous random variable it does not matter, it is turned on during this interval of its border or not:
P (α.< X < β) = P(α ≤ X < β) = P(α ≤ X ≤ β)
Distribution density continuous random variable called function
f (x) \u003d f '(x), derived from the distribution function.

Distribution density properties

1. The density of the distribution of a random variable is non-negative (F (x) ≥ 0) for all X values.
2. Normal condition:

The geometrical meaning of the normalization conditions: the area under the distribution density curve is one.
3. The probability of incoming random variance X into the gap from α to β can be calculated by the formula

The geometrically probability of contacting the continuous random variable x into the gap (α, β) is equal to the area of \u200b\u200bthe curvilinear trapezion under the distribution density curve based on this gap.
4. The distribution function is expressed through the density as follows:

The value of the distribution density at point x is not equal to the likelihood to take this value, for a continuous random variable, it can only be about the likelihood of entering the specified interval. Let if its probability density looks like:

Mathematical expectation and dispersion of uniformly distributed random variance are determined by expressions

3.8. Random value X. It is distributed evenly on the segment. Find the distribution function F.(x.), mathematical expectation, dispersion and average quadratic deviation of the magnitude.

Decision. Probability density for magnitude X. It has the form:

Consequently, the distribution function calculated by the formula:

,

wrongs as follows:

Mathematical expectation will be equal M X. \u003d (1 + 6) / 2 \u003d 3.5. We find dispersion and the average quadratic deviation:

D X. = (6 – 1) 2 /12 = 25/12, .

Normal distribution

Random value X. Distributed according to a normal law if its probability distribution density feature has the form:

where M X. - expected value;

- Average quadratic deviation.

The likelihood of random variance to the interval ( but, b.) is on the formula

R(but < X. < b.) \u003d F - F \u003d f ( z. 2) - f ( z. 1), (5)

where f ( z.) \u003d - Laplace function.

Laplace function values \u200b\u200bfor different values z. shown in Appendix 2.

3.9. Mathematical expectation of a normally distributed random variable X. equally M X. \u003d 5, dispersion is equal D X. \u003d 9. Write an expression for probability density.

3.10. Mathematical expectation and secondary quadratic deviation of a normally distributed random variable X. Accordingly, 12 and 2. Find the likelihood that the random value will take a value concluded in the interval (14; 16).

Decision. We use formula (21.2), considering that M X. = 12, = 2:

R(14 < X. < 16) = Ф((16 – 12)/2) – Ф(14 – 12)/2) = Ф(2) – Ф(1).

On the table of values \u200b\u200bof the function of the Laplace, we find F (1) \u003d 0.3413, Φ (2) \u003d 0.4772. After the substitution, we obtain the value of the desired probability:

R(14 <H. < 16) = 0,1359.

3.11. There is a random value X., distributed according to a normal law, the mathematical expectation of which is 20, the average quadratic deviation is 3. to find an interval symmetrical on the mathematical expectation in which with probability r \u003d 0.9972 Random value will fall.

Decision. As R(h. 1 < H. < h. 2) = r \u003d 2F (( h. 2 – M X.) /), then f ( z.) = r/ 2 \u003d 0.4986. On the table function of Laplas find the value z.corresponding to the value obtained function f ( z.) = 0,4986: z. \u003d 2.98. Given the fact that z. = (h. 2 – M X.) /, define \u003d h. 2 – M X. = z. \u003d 3 · 2.98 \u003d 8.94. The desired interval will be viewed (11.06; 28.94).

We take into account what f.(x.) = F "(x.). Then we get:

Substitute to the expression for mathematical expectation

.

Integrating in parts, we get M X. \u003d 1 /, or M X. = 1/0,1.

To determine the dispersion, we integrate the first components in parts. As a result, we get:

.

Take into account the expression found for M X.. From

.

In this case M X. = 10, D X. = 100.

Random Systems

The result of any random experiment can be characterized by qualitatively and quantitatively. Qualitative The result of a random experiment - random event. Any quantitative characteristicwhich as a result of a random experiment can take one of some many values \u200b\u200b- random value.Random value it is one of the central concepts of probability theory.

Let - arbitrary probabilistic space. Random variablecalled the actual numerical function x \u003d x (w), w w, such that with any valid x. .

Event it is accepted in the form of x< x.. In the future, random variables will be denoted by lowercase Greek letters X, H, Z, ...

A random variable is the number of points that have dropped when throwing a playing bone, or an increase of a chance chosen from the student's training group. In the first case we are dealing with discrete random variable (It takes values \u200b\u200bfrom a discrete numerical set. M \u003d.(1, 2, 3, 4, 5, 6); In the second case - with continuous random variable (It takes values \u200b\u200bfrom a continuous numerical set - from a numerical straight interval I.=).

Each random value is fully determined by its distribution function.

If x. - Random value, then function F.(x.) = F X.(x.) = P.(X.< x.) Called distribution functionrandom variable x. Here P.(X.< X.) - the likelihood that the random value x takes a value less x..

It is important to understand that the distribution function is a "passport" of a random variable: it contains all information about the random amount and therefore study of a random value is to study it distribution functionswhich is often called simply distribution.

The distribution function of any random variable has the following properties:

If X is a discrete random value that makes values x. 1 < X. 2 < … < X I. < … с вероятностями p. 1 < P. 2 < … < P I. < …, то таблица вида

x. 1	x. 2	…	x I.	…
p. 1	p. 2	…	p I.	…

called distribution of a discrete random variable.

The distribution function of a random variable, with such a distribution, has the form

The discrete random value of the distribution function is stepped. For example, for a random number of points that have fallen out when one throwing a playing bone, distribution, distribution function, and a distribution function schedule resemble:

1	2	3	4	5	6
1/6	1/6	1/6	1/6	1/6	1/6

If the distribution function F X.(x.) continuous, then the random value of X is called continuous random variable.

If the function of the distribution of a continuous random variable differential, then a more visual understanding of a random value gives the probability density of the random variable P x(x.), which is associated with the distribution function F X.(x.) Formulas

and .

Hence, in particular, it follows that for any random variable.

When solving practical tasks, it is often required to find a value. x.in which the distribution function F X.(x.) Random value x takes the specified value p.. It is required to solve the equation F X.(x.) = p.. Solutions such equations (corresponding values x.) in the theory of probabilities are called quantians.

Quantile X P ( p.-Quvantille, level quantile p.) random variable having distribution function F X.(x.), call a solution x P.equations F X.(x.) = p., p.(0, 1). For some p. the equation F X.(x.) = p. May have several solutions, for some - not one. This means that for the corresponding random variable, some quantils are determined ambiguously, and some quanitili do not exist.

The definitions of the distribution function of the random variable and the probability density of a continuous random variable are given. These concepts are actively used in the articles on the statistics of the site. Examples of calculating the distribution function and probability density function using MS Excel functions.

We introduce the basic concepts of statistics, without which it is impossible to explain more complex concepts.

General Aggregate and Random Value

Let us have general aggregate (Population) from n objects, each of which is inherent in a certain value of a certain numerical characteristic of H.

An example of a general population (GC) can serve as a set of weights of the same type of parts that are manufactured by a machine.

Since in mathematical statistics, any conclusion is made only on the basis of the characteristics of X (abstracting from the objects themselves), then from this point of view general aggregate It is n numbers, among which, in the general case, may be the same.

In our example, GS is just a numerical array of values \u200b\u200bof parts of parts. X - Weight of one of the details.

If we select one object from the given GS, having a characteristic x, then x is random variable. By definition, any random value It has distribution functionwhich is usually denoted by f (x).

Distribution function

Distribution function Probables random variable X Call the function f (x), the value of which at point x is equal to the probability of an event x

F (x) \u003d p (x

Let us explain on the example of our machine. Although it is assumed that our machine produces only one type of details, but it is obvious that the weight of the produced parts will be slightly different from each other. This is possible due to the fact that in the manufacture of various material could be used, and the treatment conditions could also differ slightly, etc. Let the heavier part, produced by the machine, weigh 200 g, and the easiest - 190. The likelihood that accidentally The selected part X will weigh less than 200 g equal to 1. The likelihood that will weigh less 190 g is equal to 0. Intermediate values \u200b\u200bare determined by the form of the distribution function. For example, if the process is tuned to the manufacture of parts weighing 195 g, it is reasonable to assume that the probability of choosing the part is lighter than 195 g is 0.5.

Typical graph Distribution functions For a continuous random variable, shown in the picture below (purple curve, see the example file):

In Help MS Excel Distribution function Call Integral distribution function (Cumulative.Distribution.Function., CDF.).

Let's give some properties Distribution functions:

• Distribution function F (X) varies in the interval, because Its values \u200b\u200bare equal to probabilities of the corresponding events (by definition, probability may be between 0 to 1);
• Distribution function- non-profit function;
• The likelihood that a random value took value from a certain range probability density equal to 1 / (0.5-0) \u003d 2. And for with the parameter lambda\u003d 5, value probability densityat point x \u003d 0.05 is 3.894. But, while you can make sure that the probability on any interval will be, as usual, from 0 to 1.

  Recall that distribution densityis derived from distribution functions. "Speed" of its changes: p (x) \u003d (f (x2) -f (x1)) / dx with dx tend to 0, where dx \u003d x2-x1. Those. the fact that distribution density \u003e 1 means only that the distribution function grows rather quickly (this is obviously on the example).

  Note: Square, entirely prisonered under the entire curve depicting distribution density, equal to 1.

  Note: Recall that the F (X) distribution function is called MS Excel features integral distribution function. This term is present in the parameters of the functions, for example, in norms. ARP (X; average; standard_ integral). If the MS Excel function must return Distribution functionthen parameter integral, D.B. Installed truth. If you want to calculate probability density, then parameter integral, D.B. FALSE.

  Note: For discrete distribution The probability of a random value to take some value is also often called a probability density (eng. Probability Mass Function (PMF)). In Help MS Excel probability density It may even be called the "probabilistic measure" function (see Binoma () function).

  Calculation of probability density using MS Excel functions

  It is clear that to calculate probability density For a certain value of a random variable, it is necessary to know its distribution.

  Find probability density For n (0; 1) at x \u003d 2. To do this, write a formula \u003d Norm.st.Rasp (2; false)\u003d 0.054 or \u003d Norms. Rasp (2; 0; 1; false).

  Recall that probability that continuous random amount will take a specific value x equal to 0. For continuous random variable X You can calculate only the likelihood of an event that X will take a value concluded in the interval (A; b).

  Calculation of probabilities using MS Excel functions

  1) We find the likelihood that a random variable, distributed by (see the picture above), has taken positive value. According to the property Distribution functions The probability is f (+ ∞) -f (0) \u003d 1-0.5 \u003d 0.5.

  Norm.st.Sp. (9,999E + 307; truth) -norm.st.Sp. (0; truth) =1-0,5.
  Instead of + ∞, a value of 9.99 (+ 307 \u003d 9.9999 * 10 ^ 307 was introduced into the formula, which is the maximum number that can be entered into the MS Excel cell (so to speak, the closest K + ∞).

  2) We find the likelihood that a random variable distributed by , accepted a negative value. According to the definition Distribution functions The probability is equal to f (0) \u003d 0.5.

  In MS Excel, to find this probability, use the formula \u003d Norm.st.Rasp (0; truth) =0,5.

  3) We find the likelihood that a random value distributed by standard normal distributionwill take a value concluded in the interval (0; 1). The probability is f (1) -f (0), i.e. From the probability to choose x from the interval (-∞; 1) you need to make sure the probability of choose x from the interval (-∞; 0). In MS Excel, use the formula \u003d Norm.st.Sp (1; truth) - norm.st.Sp. (0; truth).

  All calculations given above belong to a random variable distributed by standard Normal Law N (0; 1). It is clear that the probability values \u200b\u200bdepend on the specific distribution. In the article of the distribution function to find a point for which F (x) \u003d 0.5, and then find the abscissa of this point. The abscissa point \u003d 0, i.e. the probability that the random variable will take a value<0, равна 0,5.

  In MS Excel, use the formula \u003d norms. Support (0.5) \u003d 0.

  

  Definitely calculate the value random variable Allows the property of monotony distribution functions.

  Reverse distribution functioncalculates that are used, for example, when. Those. In our case, the number 0 is 0.5-quantile normal distribution. In the example file, you can calculate the other kwantil This distribution. For example, 0.8 quantil is 0.84.

  

  In English-language literature reverse distribution function It is often called PERCENT POINT FUNCTION (PPF).

  Note: When calculating quantile MS Excel Functions are used: norms.stre.Be (), Lognors. PROF (), HAY2.OB (), Gamma. Production (), etc. Learn more about the distributions presented in MS Excel, you can read in the article.

  Definition. Continuouscall a random value that can take all values \u200b\u200bfrom some finite or infinite gap.

  For a continuous random variable, the concept of the distribution function is introduced.

  Definition. Distribution function The probabilities of the random variable are called the function f (x), which determines for each value x the likelihood that the random value x will take a value of less x, that is:

  F (x) \u003d p (x< x)

  Often instead of the term "distribution function" use the term "integral distribution function".

  Distribution function properties:

  1. The values \u200b\u200bof the distribution function belong to the segment:

  0 ≤ F (x) ≤ 1.

  2. The distribution function is a non-selling function, that is:

  if x\u003e x,

  that f (x) ≥ f (x).

  3. The likelihood that a random value will take a value concluded in the interval)