P.1.3. Numerical characteristics of random variables.

The distribution function of the random variable W (X) gives its complete statistical description. However, to solve many practical problems, knowledge of only certain numerical characteristics of the distribution functions is sufficient.
The most common are the numerical characteristics given below. The mean value (mathematical expectation) (X) of a random variable X is called the sum of the products of all its variant values ​​on the probability of the occurrence of its values. If the distribution density W (X) is a continuous function, then the probability of occurrence of the value X in the interval dX is equal to W (X) dX and, then:
(x) = | Xw (x) dx.
If X takes discrete values, then:

(P.1.9)
where N is the number of possible values ​​of the random variable X, W (Xt) is the distribution density of the probability of the appearance of a random variable with the value Xt.
The most probable value (mode — mod).
The most probable value (fashion) xn

call the value of X, at which the density distribution of W (X) is maximum. The function W (X) can have one or several maxima (polymodal distributions) or not have a maximum (uniform distributions).
There are W (X) distributions having a minimum (auto-modal distributions). In the general case, the most probable value of Hnne coincides with the average value: Hn * (X). Median. Median Hm is the value of X for which the probability
/> (*,) * 0.5,
Hmmozhet does not coincide with (X) and Hyutak as P (Hm) = 0.5 = JW (X) dX.
hg
Dispersion. The variance A = a2 characterizes the scatter of the random variable X with respect to the average value. For a continuous random variable, the variance is defined as:

(A.1.11)
lgt | n
where o is the rms value of the random variable, and for discrete:
(P. 1.12)
where dX is the infinitesimal step of X.
Thus, variance has the value of the square of a random variable.
(For example, for a random variable E (t), the variance is:
N 2
A = a2 = £ (£, - (£)) - W (E,) dE, (A.1.13)
1-1
at the same time, A ~ E2 determines the power flux density (P ~ E2 (by dimension)).
Consider the following example.
Determine the variance of the random variable, which varies from Xmin = -ooo to Xtah + + °°> of a continuous random process:
D = J (A '- (Z)) 2 - ^ (Z) "! Y = ^ X2-W (X) dX + ^ (x) 2-W (X) dX -
-2] x (x) -w (x) dx = 1K (X2) - ((X)) 2),
that is, we obtain the difference between the mean of the square of the random variable and the square of the rms value.
The root-mean-square value of (or sometimes they say: standard (root-mean-square) deviation) is defined as the square root of the variance value:
about = l / d (P.1.14)
that is, the quantity a has the dimension of the random variable X and more conveniently characterizes the scatter of the values ​​of the random variable than the variance. For a discrete random value of the field strength, the value is derived from the formula (P. 1.13):
a = jz (£ .- (£)) 2'w (^) ^ "(P. 1.15)
where dE is the infinitesimal change of E.