How to find the expected value through the distribution function. Expectation of a discrete random variable

Expected value

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).

Distribution density f(x) given Distribution function F(x) given

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

1. The distribution density of the random variable is non-negative (f(x) ≥ 0) for all values ​​of x.
2. Normalization condition:

The geometric meaning of the normalization condition: the area under the distribution density curve is equal to unity.
3. The probability of a random variable X falling into the interval from α to β can be calculated using the formula

Geometrically, the probability of a continuous random variable X falling into the interval (α, β) is equal to the area of ​​the curvilinear trapezoid under the distribution density curve based on this interval.
4. The distribution function is expressed in terms of density as follows:

The value of the distribution density at point x is not equal to the probability of accepting this value; for a continuous random variable we can only talk about the probability of falling into a given interval. Let )