Probability distribution function of a random variable and its properties.

Consider the function F(x), defined on the entire number line as follows: for each X meaning F(x) is equal to the probability that a discrete random variable will take a value less than X, i.e.

(18)

This function is called probability distribution function, or briefly, distribution function.

Example 1. Find the distribution function of the random variable given in example 1, point 1.

Solution: It is clear that if , then F(x)=0, since it does not take values ​​less than one. If , then ; if , then . But the event<3 в данном случае является суммой двух несовместных событий: =1 и =2. Следовательно,

So for we have F(x)=1/3. The function values ​​in the intervals , and are calculated similarly. Finally, if x>6 That F(x)=1, since in this case any possible value (1, 2, 3, 4, 5, 6) less than x. Graph of a function F(x) shown in Fig. 4.

Example 2. Find the distribution function of the random variable given in example 2, paragraph 1.

Solution: It's obvious that

Schedule F(x) shown in Fig. 5.

Knowing the distribution function F(x), it is easy to find the probability that a random variable satisfies the inequalities.
Consider the event that a random variable will take a value less than . This event splits into the sum of two incompatible events: 1) the random variable takes values ​​less than , i.e. ; 2) the random variable takes values ​​that satisfy the inequalities. Using the addition axiom, we get

But by definition of the distribution function F(x)[cm. formula (18)], we have , ; therefore,

Thus, the probability of a discrete random variable falling into an interval is equal to the increment of the distribution function over this interval.

Let's consider the basic properties of the distribution function.
1°. The distribution function is non-decreasing.
In fact, let< . Так как вероятность любого события неотрицательна, то . Therefore, from formula (19) it follows that , i.e. .

2°. The distribution function values ​​satisfy the inequalities .
This property follows from the fact that F(x) defined as probability [see formula (18)]. It is clear that * and .

3°. The probability that a discrete random variable will take one of the possible values ​​xi is equal to the jump in the distribution function at point xi.
Indeed, let xi is the value taken by the discrete random variable, and . Assuming , , in formula (19), we obtain

Those. meaning p(xi) equal to function jump** xi. This property is clearly illustrated in Fig. 4 and fig. 5.

* Hereinafter the following notations are introduced: , .
** It can be shown that F(xi)=F(xi-0), i.e. what is the function F(x) is left continuous at a point xi.

3. Continuous random variables.

In addition to discrete random variables, the possible values ​​of which form a finite or infinite sequence of numbers that do not completely fill any interval, there are often random variables whose possible values ​​form a certain interval. An example of such a random variable is the deviation from the nominal value of a certain size of a part with a properly adjusted technological process. This kind of random variables cannot be specified using the law of probability distribution p(x). However, they can be specified using the probability distribution function F(x). This function is defined in exactly the same way as in the case of a discrete random variable:

Thus, here too the function F(x) defined on the entire number line, and its value at the point X is equal to the probability that the random variable will take a value less than X.
Formula (19) and properties 1° and 2° are valid for the distribution function of any random variable. The proof is carried out similarly to the case of a discrete quantity.
The random variable is called continuous, if for it there is a non-negative piecewise continuous function* that satisfies for any values x equality

Based on the geometric meaning of the integral as an area, we can say that the probability of fulfilling the inequalities is equal to the area of ​​a curvilinear trapezoid with a base , bounded above by the curve (Fig. 6).

Since , and based on formula (22)

Note that for a continuous random variable the distribution function F(x) continuous at any point X, where the function is continuous. This follows from the fact that F(x) is differentiable at these points.
Based on formula (23), assuming x 1 =x, , we have

Due to the continuity of the function F(x) we get that

Hence

Thus, the probability that a continuous random variable can take on any single value x is zero.
It follows that the events consisting in the fulfillment of each of the inequalities

They have the same probability, i.e.

In fact, for example,

Because

Comment. As we know, if an event is impossible, then the probability of its occurrence is zero. With the classical definition of probability, when the number of test outcomes is finite, the converse proposition also holds: if the probability of an event is zero, then the event is impossible, since in this case none of the test outcomes favors it. In the case of a continuous random variable, the number of its possible values ​​is infinite. The probability that this quantity will take on a specific value x 1 as we have seen, is equal to zero. However, it does not follow from this that this event is impossible, since as a result of the test the random variable can, in particular, take the value x 1. Therefore, in the case of a continuous random variable, it makes sense to talk about the probability of the random variable falling into the interval, and not about the probability that it will take on a specific value.
So, for example, when making a roller, we are not interested in the probability that its diameter will be equal to the nominal value. What is important to us is the probability that the diameter of the roller is within the tolerance range.

Probability distribution function and its properties.

The probability distribution function F(x) of a random variable X at point x is the probability that, as a result of an experiment, the random variable will take on a value less than x, i.e. F(x)=P(X< х}.
Let's consider the properties of the function F(x).

1. F(-∞)=lim (x→-∞) F(x)=0. Indeed, by definition, F(-∞)=P(X< -∞}. Событие (X < -∞) является невозможным событием: F(-∞)=P{X < - ∞}=p{V}=0.

2. F(∞)=lim (x→∞) F(x)=1, since by definition, F(∞)=P(X< ∞}. Событие Х < ∞ является достоверным событием. Следовательно, F(∞)=P{X < ∞}=p{U}=1.

3. The probability that a random variable will take a value from the interval [Α Β] is equal to the increment of the probability distribution function on this interval. P(Α ≤X<Β}=F(Β)-F(Α).

4. F(x 2)≥ F(x 1), if x 2, > x 1, i.e. The probability distribution function is a non-decreasing function.

5. The probability distribution function is left continuous. FΨ(x o -0)=limFΨ(x)=FΨ(x o) for x→ x o

The differences between the probability distribution functions of discrete and continuous random variables can be well illustrated with graphs. Let, for example, a discrete random variable have n possible values, the probabilities of which are equal to P(X=x k )=p k , k=1,2,..n. If x ≤ x 1, then F(X)=0, since there are no possible values ​​of the random variable to the left of x. If x 1< x ≤ x 2 , то левее х находится всего одно возможное значение, а именно, значение х 1 .

This means F(x)=P(X=x 1 )=p 1 .At x 2< x ≤ x 3 слева от х находится уже два возможных значения, поэтому F(x)=P{X=x 1 }+P{X=x 2 }=p 1 +p 2 . Рассуждая аналогично,приходим к выводу, что если х k < x≤ x k+1 , то F(x)=1, так как функция будет равна сумме вероятностей всех возможных значений, которая по условию нормировки равна еденице. Таким образом, график функции распределения дискретной случайной величины является ступенчатым. Возможные значения непрерывной величины располагаются плотно на интервале задания этой величины, что обеспечивает плавное возрастания функции распределения F(x), т.е. ее непрерывность.

Let's consider the probability of a random variable falling into the interval , Δx>0: P(x≤X< x+Δx}=F(x+ Δx)-F(x). Перейдем к пределу при Δx→0:

lim (Δx→0) P(x≤ X< x+Δx}=lim (Δx→0) F(x+Δx)-F(x). Предел равен вероятности того, что случайная величина примет значение, равное х. Если функция F(x) непрерывна в точке х, то lim (Δx→0) F(x+Δx)=F(x), т.е. P{X=x}=0.

If F(x) has a discontinuity at point x, then the probability P(X=x) will be equal to the function jump at this point. Thus, the probability of any possible value occurring for a continuous value is zero. The expression P(X=x)=0 should be understood as the limit on the probability of a random variable falling into an infinitesimal neighborhood of point x for P(Α< X≤ Β},P{Α ≤ X< Β},P{Α< X< Β},P{Α ≤ X≤ Β} равны, если Х - непрерывная случайная величина.

For discrete variables, these probabilities are not the same in the case when the boundaries of the interval Α and (or) Β coincide with the possible values ​​of the random variable. For a discrete random variable, it is necessary to strictly take into account the type of inequality in the formula P(Α ≤X<Β}=F(Β)-F(Α).

In the previous n° we introduced the distribution series as an exhaustive characteristic (distribution law) of a discontinuous random variable. However, this characteristic is not universal; it exists only for discontinuous random variables. It is easy to see that such a characteristic cannot be constructed for a continuous random variable. Indeed, a continuous random variable has an infinite number of possible values, completely filling a certain interval (the so-called “countable set”). It is impossible to create a table listing all possible values ​​of such a random variable. Moreover, as we will see later, each individual value of a continuous random variable usually does not have any nonzero probability. Consequently, for a continuous random variable there is no distribution series in the sense in which it exists for a discontinuous variable. However, different areas of possible values ​​of a random variable are still not equally probable, and for a continuous variable there is a “probability distribution,” although not in the same sense as for a discontinuous one.

To quantitatively characterize this probability distribution, it is convenient to use not the probability of the event, but the probability of the event, where is some current variable. The probability of this event obviously depends on , there is some function of . This function is called the distribution function of a random variable and is denoted by:

. (5.2.1)

The distribution function is sometimes also called the cumulative distribution function or the cumulative distribution law.

The distribution function is the most universal characteristic of a random variable. It exists for all random variables: both discontinuous and continuous. The distribution function fully characterizes a random variable from a probabilistic point of view, i.e. is one of the forms of the distribution law.

Let us formulate some general properties of the distribution function.

1. The distribution function is a non-decreasing function of its argument, i.e. at .

2. At minus infinity, the distribution function is equal to zero:.

3. At plus infinity, the distribution function is equal to one: .

Without giving a rigorous proof of these properties, we will illustrate them using a visual geometric interpretation. To do this, we will consider a random variable as a random point on the Ox axis (Fig. 5.2.1), which as a result of experiment can take one position or another. Then the distribution function is the probability that a random point as a result of experiment will fall to the left of point .

We will increase , that is, move the point to the right along the abscissa axis. Obviously, in this case, the probability that a random point will fall to the left cannot decrease; therefore, the distribution function cannot decrease with increasing.

To make sure that , we will move the point to the left along the abscissa indefinitely. In this case, hitting a random point to the left in the limit becomes an impossible event; It is natural to believe that the probability of this event tends to zero, i.e. .

In a similar way, by moving the point to the right indefinitely, we make sure that , since the event becomes, in the limit, reliable.

The graph of the distribution function in the general case is a graph of a non-decreasing function (Fig. 5.2.2), the values ​​of which start from 0 and reach 1, and at certain points the function may have jumps (discontinuities).

Knowing the distribution series of a discontinuous random variable, one can easily construct the distribution function of this variable. Really,

,

where the inequality under the sum sign indicates that the summation applies to all those values ​​that are less than .

When the current variable passes through any of the possible values ​​of the discontinuous value, the distribution function changes abruptly, and the magnitude of the jump is equal to the probability of this value.

Example 1. One experiment is performed in which the event may or may not appear. The probability of the event is 0.3. Random variable – the number of occurrences of an event in an experiment (characteristic random variable of an event). Construct its distribution function.

Solution. The value distribution series has the form:

Let's construct the distribution function of the value:

The distribution function graph is shown in Fig. 5.2.3. At discontinuity points, the function takes on the values ​​marked with dots in the drawing (the function is continuous on the left).

Example 2. Under the conditions of the previous example, 4 independent experiments are performed. Construct a distribution function for the number of occurrences of an event.

Solution. Let us denote the number of occurrences of the event in four experiments. This quantity has a distribution series

Let's construct the distribution function of a random variable:

3) at ;

In practice, usually the distribution function of a continuous random variable is a function that is continuous at all points, as shown in Fig. 5.2.6. However, it is possible to construct examples of random variables, the possible values ​​of which continuously fill a certain interval, but for which the distribution function is not continuous everywhere, but suffers a discontinuity at certain points (Fig. 5.2.7).

Such random variables are called mixed. An example of a mixed value is the area of ​​destruction caused to a target by a bomb, the radius of destructive action of which is equal to R (Fig. 5.2.8).

The values ​​of this random variable continuously fill the interval from 0 to , occurring at bomb positions of types I and II, have a certain finite probability, and these values ​​correspond to jumps in the distribution function, while in intermediate values ​​(position of type III) the distribution function is continuous. Another example of a mixed random variable is the failure-free operation time T of a device tested for time t. The distribution function of this random variable is continuous everywhere except point t.

Dispersion continuous random variable X, the possible values ​​of which belong to the entire Ox axis, is determined by the equality:

Purpose of the service. The online calculator is designed to solve problems in which either distribution density f(x) or distribution function F(x) (see example). Usually in such tasks you need to find mathematical expectation, standard deviation, plot functions f(x) and F(x).

Instructions. Select the type of source data: distribution density f(x) or distribution function F(x).

The distribution density f(x) is given:

The distribution function F(x) is given:

A continuous random variable is specified by a probability density
(Rayleigh distribution law - used in radio engineering). Find M(x) , D(x) .

The random variable X is called continuous , if its distribution function F(X)=P(X< x) непрерывна и имеет производную.
The distribution function of a continuous random variable is used to calculate the probability of a random variable falling into a given interval:
P(α< X < β)=F(β) - F(α)
Moreover, for a continuous random variable, it does not matter whether its boundaries are included in this interval or not:
P(α< X < β) = P(α ≤ X < β) = P(α ≤ X ≤ β)
Distribution density a continuous random variable is called a function
f(x)=F’(x) , derivative of the distribution function.

Properties of distribution density