14. Рівномірний закон розподілу.

Uniform law of distribution.

RV X is called uniformly distributed in interval [a;b],

if it’s density of distribution looks like:

f (x)= C, x є [a;b],

0, x ¢ [a;b],

We can find const applying a properties of differential function:

_+∞

∫ f(x)dx=1

^-∞

_{+∞ a b +∞ b}

∫f(x)dx=∫f(x)dx + ∫f(x)dx+ ∫f(x)dx=∫cdx=cx│^b_a=c(b-a)=1

^{-∞ -∞ a b a}

Consequently

f(x)= b-a, x є [a;b],

0, x ¢ [a;b],

S =1/(b-a)*(b-a)

For uniform distribution integral function is following:

F (x)= 0; x<a

x-a; a≤x≤b;

b-a

1; x>b

Numerical characteristics for uniform distribution

_{+∞ a b +∞ b}

M(X) = ∫xf(x)dx=∫0dx + ∫1/(b-a)xdx+∫0dx=1/(b-a)*x²/2│=

^{-∞ -∞ a b a}

⁼ b² – a²­­__ = (b-a)(b+a) = a+b

2(b-a) 2(b-a) 2(b-a) 2

D(X)=M(X²)-M²(X)= a²+ab+b² _ (a+b)² = (b-a)²

3 2 12

δ(X)= b-a

2√3

The graph of integral distribution function for uniform

d istribution is following

F(x)= 0,x<a;

(x-a)/(b-a), a≤x≤b;

1, x>b

15. Показниковий закон розподілу

Exponential law of distribution.

CRV X is called distributed by exponential law with parameter λ>0, if its differential distribution function looks like:

f (x)= 0,x<0;

λe^-λx, x≥0

_+∞

∫f(x)dx=1

^-∞

The graph of integral distribution function for exponential distribution is following:

F (X)= 0,x<0;

1-e^-λx, x≥0

Numerical characteristics:

1. M(X)=1/λ;

2. D(X)=1/λ²;

3. δ(X)= (b-a)/2√3.

16. Нормальний закон розподілу. Характерні ознаки нормально розподіленої випадкової величини.

Normal law of distribution.

RV X is said to be distrebuted by normal law, if it’s density of distribution of differential function is following:

, where a and δ are parameters of distribution, δ>0.

N umerical characteristics of the normal law:

[(x-a)/δ=t and apply Poisson’s

_+∞

integral]= ∫e^-t2/2=√2п

^-∞

Finally we obtain M(X)=a

D(X)=M(X²)-M²(X)=δ²

δ(X)=δ

Properties of differential function of normal distribution:

1. f(x) is certain for x є R, and f(x) is continous for x

2. f(x)>0 for x;

3. f(-x)=f(x); x=a

The graph of the function f(x) is symmetric considering with straight line x=a;

4. The point of f_max is f(a) (f_max = f(a));

5. lim f(x)=0

^x->±∞

Integral function of normal distribution

_x

F(x)=∫f(t)dt

^-∞

_x

F(x) = 1 * ∫e^{-(t-a)² /2δ²} dt=[(t-a)/δ=z and apply Poisson’s integral]

δ√2п ^-∞ _x

=[Let’s denote: Ф(x)=1/√2п *∫e ^-t²/2dt]

⁰

Finally we obtain: F(x)=1/2+Ф((x-a)/δ)

Properties of integral function of Laplace:

1. Function Ф(X) is certain and continuous for x є R;

2. Ф (0)=0

3. Function Ф(x) is add: Ф(-х)=Ф(х);

4. Lim Ф(x) = ±1/2

^x->±∞

^x-a