Dispersion characterizes on average degree of dispersion of SV values concerning its average value, that is shows, values X around mx are how densely grouped. If SV has dimension (it can be expressed in any units), then the dimension of dispersion is equal to a square of dimension St.
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Instruction
1. For consideration of the matter it is necessary to enter some designations. Exponentiation will be designated through a symbol "^", a root square – "sqrt", and designations which concern integrals are given in fig. 1.
2. Let the average value (expected value) mx of the random variable (RV) H. be known it is necessary to remind that operator designation expected value mx = M {X} = M [X], at the same time for it is fair property M {aX} = aM {X}. Expected value of a constant is this constant (M {a } =a). Besides, it is necessary to enter a concept of aligned St. Hts = H-mx. Obviously, M {HC} = M {X} – mx=0
3. Dispersion of SV (Dx) call the expected value of a square aligned St. Dx =int ((X-^2) W (x) dx). At this W(x) - density of probability St. For discrete SV Dx = (1/n) ((X-^2 + (x2 mx) ^2+ …+ (xn-mx) ^2). For dispersion, as well as for expected value, the operator record Dx = D [X] is provided (or D {X}).
4. From definition of dispersion follows that the same way it can be found on the following formula: Dx=M {(X-^2 } =D {X} = M {Xц^2}. In practice as an example of characteristic of dispersion use an average square of a deviation of SV more often (SKO - a mean square deviation). bkh =sqrt(Dx), at the same time the dimension of X and SKO coincide [X]= [bkh].
5. Properties of dispersion.1. D [a]=0. Really, D [a] = M [(a-a) ^2] =0 (the physical sense - at a constant is not present dispersion).2. The D [aX]= (a^2) D [X], as M {(aX-M[aX])^2} = M {(aX - (amx)) ^2 } = (a^2) of M {(X - mx) ^2 } = (a^2) D {X}.3. Dx = M {X^2} (-mx^2), since M {(X - mx) ^2 } = M {X^2 - 2Xmx + mx^2 } = M {X2} - 2M {X} mx + mx2 == M {X^2} - 2mx^2+mx^2=M{X^2} – mx^2.4. If CB X and Y are independent, then M {XY} = M {X} M {Y}.5. D {X+Y} =D {X-Y } =D {X} + D {Y}. Really, considering that X and Y are independent, also Hts and Yts are independent. Then, for example, D {X-Y }=M {((X-Y)-M[X-Y]) ^2 } = M {((X-(Y-my)) ^2 } = M {Xц^2} + M {Yц^2} - M {Xц^2} M {Yц^2 } =DxDy.
6. Example. Density of probability of accidental tension of X (cm of fig. 2) is given. To find its dispersion and SKO. Decision. On a probability density normalization condition, the area under the schedule of W (x) is equal to 1. As it is a triangle, (1/2)4W(4)=1. Then W(4)=0,5 1/B. From here W (x)= (1/8) x. mx=int (0 - 4) (x (x/8) dx == (x^3)/24 | (0 – 4) =8/3. At calculation of dispersion is the most convenient to use its 3rd property: Dx = the M {X^2} (-mx^2) =int (0 - 4) ((x^2) (x | 8) dx - 64 | 9= (x^4)/32) | (0 – 4)-64/9 = 8-64/9=8/9.