The function (1) has the following properties

PROBABILITY SPACE

Theorem 1. The function (1) has the following properties

F1. If x₁x₂ then F x₁ dF x₂ (i.е. F F x( ) is a monotonically nondecreasing function);

F2. lim ( ) 0

F x F x

f p f , lim ( ) 1

F x F x

f n f ;

F3. F x( ) _{is a} right-continuous function

^{F x( 0) F x( ) ,x R}

, it has left limits at each point xR.

86

Proof. The property F1 is the corollary of the following. As

@ @

1 , 1 , 2 2

x x

A f x Ž f x A , then by the property of probability P A

x1 dP Ax2 . The property F2 is the corollary of the following. From x_np f it implies that

f^, xn

@

p ‡, from y_nn f it implies that

f^, yn

@

nR. We also use a continuity from below (property Р3″) and continuity from above (property Р3′) of probability and monotonicity of the function ( )F x .

It is not difficult to see that property F3 is also a consequence of the properties of continuity from below and from above. ז Definition 1. The function F F x( ) satisfying properties F1, F2, F3 is called the distribution function on the number line R.

Thus, by the Theorem 1, the distribution function F F x( ), defined by (1), corresponds to each probability function P in the space

^{R, ( )E R}

. It turns out that the converse is also true.

Theorem 2. Let the function F F x( ) be the distribution function on the number scale R f f( , ).

Then on

^{R, ( )E R}

there is the only one probabilistic measure P such that for any interval f d fa b the following takes place:

^,

@

^{( ) ( )}

P a b F b F a . (2) Proof. By the theorem on the extension of the probability, for constructing a probability space

R,E R ,P

, it suffices to specify the probability P on the algebra

ࣛ generated by intervals of the form

a,b

@

(because ߪሺࣛሻ ൌ ߚሺܴሻ). But we know that any element A of algebra ࣛ can be written in the form of a finite sum of disjoint intervals of the form

a,b

@

:

@

1

, ,

n

i i i i

i

A

¦

a b a b^. (a_i, b_i may be infinite). Let’s by definition

0 1

n

i i

i

P A

¦

ª¬F b F a º¼^.

Then the properties F1, F2 are satisfied, therefore the axioms Р1, Р2 are also fulfilled. Now it remains only to verify the countable additivity (or continuity) of ܲ_଴ on ࣛ.

Let ܤ_௡ א ࣛ, B_n1ŽB_n, ^fB B n n

1

ࣛ.

87

Let’s show that P B_{0 n} oP B₀ , when ݊ ՜ λ (property of continuity «from below»).

Without loss of generality, we can assume that B consists of only one interval

^{a b,}

@

^{: B}

^{a b,}

@

^{. As} BŽBn, then for any n there is an interval

a bn^, n

@

, such that

a bn^, n

@

ŽBn, and such that B

a b^,

@

Ž

a bn^, n

@

. Let a semi-interval

a bn^, n

@

be contained in B_n and be a maximal half-interval containing B. Furthemore, B_n, starting with some n, does not contain any half-intervals outside

a bn^, n

@

(if

c d^,

@

ŽBn for all n, then

^{c d,}

@

^ŽB). Thus, from monotonicity of B_nand the fact that ^B

^{a b,}

@

^it

follows, that starting from some number, for all n, Bn

a bn^, n

@

, where ܽ_௡ ൌ ܽǡ ܾ_௡ ՝ ܾ. But by the property F3 ,

0 n n n

P B F b F a oF b F a .

Thus, axiom P3 is also fulfilled. ז Thus, there exists a one-to-one correspondence between the probability functions

P defined on

^{R, ( )E R}

and the distribution functions F F x( ) that are defined on the number scale and satisfy the conditions F1, F2, F3.

The probability measure P constructed by the distribution function F F x( )is usually called the Lebesgue-Stiltes probability measure. Particularly important is the case when

0 , 0,

( ) , 0 1,

1, 1

x

F x x x

x

­

° d d

®° !

¯

In this case, the probability measure corresponding to F(x) and denoted by λ is called the Lebesgue measure on the segment

> @

0,1 . It is clear that Oሺܽǡ ܾ༱ ൌ ܾ െ ܽ.

In other words, the Lebesgue measure of the interval

^{a b,}

@

, as well as of any of the intervals a b, ,

^{a b,}

@

^,

>

^{a b,}

^,

> @

^{a b,} is just its length ܾ െ ܽ :

^{a b,}

@

^{a b,}

>

^{a b,}

> @

^{a b, b a}

O O O O .

According to condition F3, the distribution function F F x( ) can have only points of discontinuity of the first kind. The value

( 0) ( 0) ( ) ( 0)

px F x F x F x F x

88

is called a jump of the distribution function ( )F x at the point x. A jump of the distribu- tion function is equal to zero at points of continuity and is strictly greater than zero at the points of discontinuity.

If for any H !0, the inequality (F x H) F x( H) 0! takes place then such a point x R is called a point of growth of the distribution function ( )F x .

Theorem 3. Any distribution function F F x( ) has at most a countable number of discontinuity points.

Proof. Denote by ( )C F (here and everywhere below) the set of points of con- tinuity of F F x( ), by C(F) – the complementary set (the set of points of discon- tinuity), and by D_k – the set of points of discontinuity, in which the jumps of the func- tion ( )F x lie in the half-open interval ¹ ,¹ , 1,2,...

1 k

k k

§ º

¨ »

© ¼ . So, by definition:

^ `

( ) : ( ) ( 0) , ( ) \ ( )

C F x R F x F x C F R C F ,

1 1

\ ( ) : ( ) ( 0) ,

k x 1

D x R C F F x F x p

k k

­  § º½

® ¨© »¼¾

¯ ¿, k 1,2,... .

Then the number of elements of each set D_k is at most k, therefore the number of elements of the set

1

( ) \ ( ) _k

k

C F R C F ^f DD_kk (as a countable sum of not more than a countable number of sets) is at most countable. ז Remark 1. We obtain from the results of the theorem we have proved that the set ( )C F is everywhere dense on the real line R. But it turns out that C F can also be a countable everywhere dense set on R.

Example.

Let

^

r r1, ,...2

`

be a set of rational numbers on a line (it is well known that this set is everywhere dense on R). To each point r_k we put in correspondence a jump

k 2

k

pr . Then the set of points of discontinuity of a function

:

( ) k

k

k r x r

F x p

¦

d ^{is the}

set C F

^

r r1, ,...2

`

, and its closure is ª¬C F º ¼ R.

The fact that the function defined in this way is a distribution function is verified directly.

A distribution function F x that changes its values only at points of a finite or countable set X

^

x x1, ,...2

`

is called a discrete distribution function.

If the interval

^{a bj, j}º¼ contains only one point x_jC F( ), then

^{j, j ( )j ( )j}

P a b º ¼ F b F a . If a_jnx_j , b_j px_j , then

89

^ `

^{( ) ( 0) ,}

^ `

¹

j j j j j

j

p P x F x F x

¦

P x ^.

It is obvious that the introduced discrete distribution function F x can be rep- resented in the form

: _j j

j x x

F x p

¦

d ^.

The set of numbers p p₁, ₂,..., where ^{pj P x}

^ `

^{j , j 1}

j

¦

p , is called the dis- crete probability distribution.

Examples of frequently occurring discrete probability distributions are given in Table 1 below.

Table 1 Discrete distributions

Distribution Name Probabilities p_k Parameters

Discrete uniform k N

N1, 1,2,... . N 1,2,....

Bernoulli p₁ p, p₀ q. 0dpd1, q 1p.

Binomial C_n^kp^kq^nk,k 0,1,2,...,n. 0d pd1,q 1p,n 1,2,...

Poisson , 0,1,2,...

!

k

e _O Ok^k

. O!0.

Geometric q^k1p , k 1,2,.... 0pd1, q 1p.

Negative binomial

(Pascal) C_k^r₁¹p^rq^kr,k r,r1,... 0 pd1, q 1p. Hypergeometric ,k 0,1,...min(n,M).

C C C

Nn k

n M

k N

M

N, M, n are positive integers,

n N M

N ! , !

If there is an amnonnegative function ( )f x

^{f x t0}

such that for all x R we can write down the distribution function F x as an integral

x

F x f u du

f

³

. (3) Then such a distribution function is called an absolutely continuous distribution function.

90

The integral (3), in the general case, must be understood in the sense of the Le- besgue integral ([9]), but for our purposes (in this textbook) this integral is sufficiently understood as an (improper) Riemann integral.

From the definition of the distribution functions we obtain that any nonnegative function ( )f x , which is integrable by Riemann integral and such that

( ) 1

f x dx

f f

³

determines by formula (3) a distribution function F F x( ).

In Table 2 we give examples of different distribution densities f x( ) that are especially important for probability theory and mathematical statistics, with indicating their names and parameters.

Table 2 Absolutely continuous distributions

Distribution Name Density function f(x) Parameters

Uniform on

> @

^a,b b a ^ad^xd^b

1 ,

,

> @

ab x , ,

0  .

b a R b

a,  , .

Laplace ^V

V

a x

e

2

1 _, fxf. a,VR, V !0.

Normal or Gaussian

f f

x e

a x

2 ,

1 2

2

2V

V

S ^{. a,VR, V !0.}

Gamma

0

1 , !

e x

x^{O Tx}

O

O T

Г ^,

0 ,

0 x .

, 0 , ,O T! T R

!0 O . Beta

௫^ೝషభሺଵି௫ሻ^ೞషభ

஻ሺ௥ǡ௦ሻ ǡ Ͳ ൑ ݔ ൑ ͳ,

Ͳǡ ݔ ב ሺͲǡͳሻ ݎ ൐ Ͳǡ ݏ ൐ Ͳ.

Cochi S[T²xa²]

T _, fxf.

0 ,

,  T!

T a R . Chi-square with n

freedom degrees, F_n²

0 , 2 2

1 ₂ ¹ ₂

2

t

¸¹

¨ ·

©

§

e x x

n

x n n

Г

,

0 ,

0 x .

,...

2 , 1

n .

Стьюдента с n степенями свободы, t

2 1 2

1 2 2 1 1

¸¸¹

·

¨¨©

§

¸¹

¨ ·

©

§

¸¹

¨ ·

©

§ n

n x n

n

n Г Г

S ^,

f f

x .

,...

2 , 1

n .

91 F, F(m, n),

Fisher-Snedekor

2 2 1 2

2 1 2

n m m m

n mx x n B m

n m

¸¹

¨ ·

©§

˜

¸¹

¨ ·

©

§

¸¹

¨ ·

©

§

,

, xt0,

0

0, x .

,...

2 , 1 ,m

n .

Exponential Oe^Ox , xt0,

0 ,

0 x .

!0 O . Two-sided exponential Oe^Ox

2 ^, fxf. O!0.

In Table 2, Г( ),x B x y( , ) are (respectively) the gamma function and the beta function:

1 0

( ) ^{t x}

Г x ^f

³

e t dt ^,

1 1 1

0

( ) ( )

( , ) (1 )

( )

x y Г x Г y

B x y t t dt

Г x y

³

^.

From Table 2, we note the following: if O 1, then gamma-distribution is expo- nential; if ¹,

2 2

T O n (n is an integer), then gamma-distribution is F_n² – distribution.

In the probability theory, beta-distribution with the parameters 1

p q 2 is cal- led the law of arcsine.

It turns out that there is a third type of distribution function – this is the so-called singular distribution function. This type is characterized by the fact that the distribution function F F x( ) is continuous, but the growth point forms a set of Lebesgue mea- sure zero.

Thus, a singular distribution function F x is continuous, but F xc 0 almost everywhere, and F f F f 1.

An example of a singular distribution function is the Cantor function.

Cantor function. This function is constructed as follows:

0, 0; 1, 1

F x xd F x xt .

For ^x

> @

^0,1 define this function as follows. First, the segment

> @

0,1 is divided into three identical parts:

>

0,1/ 3 , 1/ 3, 2 / 3 , 2 / 3,1

@ > @ > @

. On the middle segment, we assume F x 1/ 2. The remaining two extreme segments are again divided into three equal parts each and on the inner segments we assume that F x 1/ 4 and

92

3 / 4

F x (respectively). Further, each of the remaining segments is again divided into three equal parts and on the inner segments we define F x as a constant which is equal to the arithmetic mean between the adjacent, already defined values of F x , etc. At points that do not belong to such internal segments, we define F x by conti- nuity. It is not difficult to see that the total length of the «internal» segments, on which

x

F is constant, is equal to

1 2 4 1 2 2 2 1 1

... 1 ... 1

3 9 27 3 3 3 3 1 2 / 3

§ § · ·

¨¨© ¨ ¸© ¹ ¸¸¹ ˜ ,

so that the function F x grows on a set of measure 0 (zero), but without jumps.

Remark 2. The question arises: is it possible to find the types of distribution