10. Дисперсія дискретної випадкової величини та її властивості. Довести (на вибір) дві з них.

Dispersion of DRV and its properties.

Dispersion of DRV is the value, what can be found by formula: D(X)=M(X-M(X))²

Theorem: Dispersion of DRV X can be evaluated by formula D(X)=M(X²)-M²(X)

Proof: by definition

D(X)=M(X²-2X*M(X)+M²(X))=M(X²)-M(2XM(X)+M(M²(X))=M(X²)=M(X²)-2M(X)*M(X)+M²(X)=M(X²)-M²(X)

Properties of dispersion:

1. D(C)=0

Proof: D(C)=M(C²)-M²(C)=C²-C²=0

2. D(CX)=C²D(X)

Proof: D(CX)=M(C²X²)-M²(CX)=C²M(X²)-M(CX)*M(CX)+C²M(X²)-CM(X)*CM(X)=C²M(X²)-C²M²(X)=C²(M(X²)-M²(X))=C²D(X)

3. D(X+Y)=D(X)+D(Y), X and Y are independent

D(X-Y)=D(X)+D(Y)

4. For independent DRV X :

D(X*Y)=M(X²)*M(Y²)-M²(X)*M²(Y)

11. Метод моментів для обчислення числових характеристик випадкових величин.

Method of moments

Let values x_i of DRV X are given with uniform step: h=x_i - _xi-1

x₀ – “conditional” zero, the value cloth to the average arithmetic.

Let’s consider RRV U: x-x₀

h

DRV U can accept only integer value. Numerical characteristics of X and U are connected by formula:

• M(X)=x₀+h*M(U)

• D(X)=h²*D(U)

• δ(X)=h*δ(U)

12. Інтегральна функція розподілу та її властивості.

Integral function of distribution and it’s properties.

The probability RV x will get a value less than x is called integral function of distribution and denoted by

F(x)=P(X<x)=p(-∞<X<x)

Considering this definition it is possible to give the definition of CRV and DRV.

1. RV is called continuous, if it’s integral function of distribution is continuous.

2. RV is called discrete, if it’s integral function of distribution is discontinuous (piecewise constant).

For DRV X, which can get the set of values: x₁,x₂,…,x_n integral function of distribution can be defined as following:

F(X)=P(X<x)=∑(X=x_j)

^xj<x

Properties of integral distribution function

1. F(X) can get values from interval 0<=F(x)<=1

2. F(-∞)=0

3. F(+∞)=1

4. F(x) – non-decreasing function

x₁,x₂ for any x₁,x₂; x₁<x₂ F(x₁)≤F(x₂)

Proof: P(X<x₂)=P(x₁≤X<x₂)+P(X<x₁)=P(x₁≤x<x₂)+F(x₁)

F(x₂)-F(x₁)=P(x₁≤X≤x₂)

F(x₂)-F(x₁)≥0

F(x₂)≥F(x₁)

Consuming:

The probability, that values of RV X Belong to the interval [x1,x2] is possible to obtain by formula:

P(x₁≤X≤x₂)=F(x₂)-F(x₁).

Corollary:

Considerate properties of integral distribution function allow to make following conclusions:

1. Any function which satisfied this condition can be integral distribution function of some RV X and opposite is also correct.

13. Диференціальна функція розподілу та її властивості.

Differential function of distribution and it’s properties.

The derivative of integral distribution function (if it exist) is called differential function of distribution and denoted by :

f(x)=F’(x)=lim ∆F

^∆x->0 ∆x

For the given function F(X) its derivative is equal to the limit of increment of function divided by an increment of argument, when the increment of argument tens to zero.

F’(X)= lim ∆F = lim F(x+∆x)-F(x)

^∆x->0 ∆x ^∆x->0 ∆x

Properties of differential function of distribution.

1. f(x)≥0

(F(x) is non-decreasing=> F’(x)=f(x)≥0)

2. The probability, that CRV X will get value from interval (a;b) is obtained by formula:

_b

P(a<X<b)=∫f(x)dx

^a

Proof: by definition f(x)=F’(x)=>F(x) is anti-derivative of f(x).

By properties of integral distribution function

_{b b}

P(a≤x≤b)=F(b)-F(a)= ∫F’(x)dx=)=∫f(x)dx.

^{a a}

3. Integral distribution function and differential distribution function are connected by formula:

_x

F(x)= ∫f(t)dt

^-∞

Corollary: condition of normalization for CRV is following

_+∞

∫f(x)dx=1

^-∞

Numerical characteristics for CRV

_+∞

M(X)= ∫f(x)dx=1

^-∞

_+∞

D(X) = ∫ [X-M(X)]²f(x)dx= M(X²)-M²(X)

^-∞