Lecture 4: Random Variables1. Definition of Random variables

1.1 Measurable functions and random variables1.2 Reduction of the measurability condition1.3 Transformation of random variables1.4 σ-algebra generated by a random variable

2. Distribution functions of random variables

2.1 Measure generated by a random variable2.2 Distribution function of a random variable2.3 Properties of distribution functions2.4 Random variables with a given distribution function2.5 Distribution functions of transformed random variables

3. Types of distribution functions

3.1 Discrete distribution functions3.2 Absolutely continuous distribution functions3.3 Singular distribution functions3.4 Decomposition representation for distribution functions

4 Multivariate random variables (random vectors)

4.1 Random variables with values in a measurable space4.2 Random vectors4.3 Multivariate distribution functions

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5 Independent random variables

5.1 Independent random variables5.2 Mutually independent random variables

1. Definition of Random variables

1.1 Measurable functions and random variables

< Y ,BY > and < X ,BX > are two measurable spaces (spaceplus σ-algebra of measurable subsets of this space);f = f(y) : Y → X is a function acting from Y to X .

Definition 4.1, f = f(y) is a measurable function if

y : f(y) ∈ A ∈ BY , A ∈ BX .

Example

(1) Y = y1, . . . , is a finite or countable set; BY is the σ-algebraof all subsets of Y . In this case, any function f(x) acting fromY to X is measurable.

(2) Y = X = R1; BY = BX = B1 is a Borel σ-algebra. In thiscase, f(x) is called a Borel function.

(3) A continuous function f = f(x) : R1 → R1 is a Borel func-tion.

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< Ω,F , P > is a probability space;X,BX is a measurable space.X = X(ω) : Ω→ X (a function acting from Ω to X ).

Definition 4.2. X = X(ω) is a random variable with valuesin space X defined on a probability space < Ω,F , P > if it is ameasurable function acting from Ω→ X , i.e., such function that

ω : X(ω) ∈ A ∈ F , A ∈ BX .

< Ω,F , P > is a probability space;X = X(ω) : Ω→ R1 (a function acting from Ω to R1);BX = B1 is a Borel σ-algebra of subsets of R1.

Definition 4.3. X = X(ω) is a (real-valued) random variabledefined on a probability space < Ω,F , P > if it is a measurablefunction acting from Ω→ R1, i.e., such function that

ω : X(ω) ∈ A ∈ F , A ∈ B1.

< Ω,F , P > is a probability space;X = X(ω) : Ω→ [−∞,+∞];B+1 is a Borel σ-algebra of subsets of [−∞,+∞] (minimal σ-

algebra containing all intervals [a, b],−∞ ≤ a ≤ b ≤ +∞);

Definition 4.4. X = X(ω) is a (improper) random variabledefined on a probability space < Ω,F , P > if it is a measurablefunction acting from Ω→ R1, i.e., such function that

ω : X(ω) ∈ A ∈ F , A ∈ B+1 .

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Examples

(1) X = x1, . . . , xN, BX is a σ-algebra of all subsets of X; Xis a random variable with a finite set of values;

(2) X = R1,BX = B1; X is a (real-valued) random variable;

(3) X = Rk,BX = Bk; X is a random vector (random variablewith values in Rk;

(4) X is a metric space, BX is a Borel σ-algebra of subsets ofX; X is a random variable with values in the metric space X.

(5) Ω = ω1, ω2 . . . → a discrete sample space;F = F0 → the σ-algebra of all subsets of Ω;

Any function X = X(ω) : Ω → R1 is a random variable since,in this case, it is automatically a measurable function.

(6) Ω = R1;F = B1 → Borel σ-algebra of subsets of R1;

1.2 Reduction of the measurability condition

The following notations are used:

X−1(A) = X ∈ A = ω : X(ω) ∈ A.

Theorem 4.1. The measurability condition (A) X−1(A) ∈F , A ∈ B1 hold if and only if (B) Ax = ω : X(ω) ≤ x ∈F , x ∈ R1.

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———————————(A) ⇒ (B). Indeed, (−∞, x], x ∈ R1 are Borel sets;

(B)⇒ (A). Indeed, let K be a class of all sets A ⊂ R1 such thatX−1(A) ∈ F . Then

(a) X−1((a, b]) = Ab \ Aa ∈ F , a ≤ b. Thus, (a, b] ∈ K, a < b;

(b) A ∈ K ⇒ A ∈ K since X−1(A) = X−1(A) ∈ F ;

(c) A1, A2 . . . ∈ K ⇒ ∪nAn ∈ K since X−1(∪nAn) = ∪nX−1(An)∈ F ;

(d) Thus K is a σ-algebra which contains all intervals. There-fore, B1 ⊆ K.———————————

1.3 Transformation of random variables

X1, . . . , Xk → random variables defined on a probability space< Ω,F , P >;f(x1, . . . , xk) : Rk → R1 → a Borel function, i.e.,

f−1(A) = (x1, . . . , xk) ∈ A ∈ Bk, A ∈ B1.

Theorem 4.2. X = f(X1, . . . , Xk) is a random variable.

———————————(a) ω : X1(ω) ∈ (a1, b1], . . . , Xk(ω) ∈ (ak, bk] ∈ F , ai ≤ bi, i =1, . . . , k;

(b) Let K be a class of all sets A ⊂ Rk such that(X1(ω), . . . , Xk(ω)) ∈ A ∈ F . Then K is a σ-algebra. The

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proof is analogous to those given for Theorem 1.

(c) C ∈ B1 ⇒ ω : f(X1(ω), . . . , Xk(ω)) ∈ C= ω : (X1(ω), . . . , Xk(ω)) ∈ f−1(C) ∈ F .

(d) Thus, X = f(X1, . . . , Xk) is a random variable.———————————

(1) U± = X1 ± X2, V = X1X2, W = X1/X2 (if X2 6= 0) arerandom variables;(2) Z+ = max(X1, . . . , Xk), Z− = min(X1, . . . , Xk) are randomvariables.

X,X1, X2, . . . , are random variables defined on a probabilityspace < Ω,F , P >;

Theorem 4.3. Let X be a random variable defining by one ofthe relation,

X = supn≥1

Xn

X = infn≥1

Xn

X = limn→∞Xn = infn≥1

supk≥n

Xk

X = limn→∞Xn = supn≥1

infk≥n

Xk

X = limn→∞Xn = limn→∞Xn = limn→∞Xn

Then X is a random variable (possibly improper).

———————————(1) ω : supn≥1Xn(ω) > x = ∪n≥1Xn(ω) > x;(2) ω : infn≥1Xn(ω) < x = ∪n≥1Xn(ω) < x;

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(3) ω : limn→∞Xn(ω) < x = ∪l≥1 ∪n≥1 ∩k≥nXk(ω) < x− 1l ;

(4) ω : limn→∞Xn(ω) > x = ∪l≥1 ∪n≥1 ∩k≥nXk(ω) > x− 1l ;

———————————

Let A1, . . . , An ∈ F and a1, . . . , an are real numbers.

Definition 4.5. X(ω) =∑n

k=1 akIAk(ω) is a simple random vari-

able.

Theorem 4.4. X = X(ω) is a random variable if and only ifX(ω) = limn→∞Xn(ω), ω ∈ Ω, where Xn, n = 1, 2, . . . are simplerandom variables.

< Z,BZ >, < Y ,BY > and < X ,BX > and are three measur-able spaces;f = f(z) : Z → Y is a measurable function acting from Z to Y .g = g(y) : Y → X is a measurable function acting from Y to X .

Theorem 4.5. The superposition h(z) = g(f(x)) of two mea-surable functions f and g is a measurable function acting fromspace Z to space X .

———————————(1) Let A ⊆ X . Then h−1(A) = f−1(g−1(A)).(2) Let A ∈ BX . Then g−1(A) ∈ BY ;(3) Then h−1(A) = f−1(g−1(A)) ∈ BZ .———————————

1.4 σ-algebra generated by a random variable

Theorem 4.6 Let X = X(ω) be a random variable defined

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on a probability space < Ω,F , P >. The class of sets FX =<X−1(A), A ∈ B1 > is a σ-algebra (generated by the random vari-able X).

———————————(a) C ∈ FX ⇔ C = X−1(A), where A ∈ B1 ⇒ C = X−1(A) =X−1(A) ∈ FX since A ∈ B1;(b) C1, C2, . . . ∈ FX ⇔ Cn = X−1(An), n = 1, 2, . . ., where An ∈B1, n = 1, 2, . . . ⇒ ∪nCn = ∪nX−1(An) = X−1(∪nAn) ∈ FX

since ∪nAn ∈ B1;(d) Thus FX is a σ-algebra.———————————(1) FX ⊆ F .

Theorem 4.7 Let X = X(ω) be a random variable definedon a probability space < Ω,F , P > and taking values in aspace X with σ-algebra of measurable sets BX . The class ofsets FX =< X−1(A), A ∈ BX > is a σ-algebra (generated by therandom variable X).

2. Distribution functions of random variables

2.1 Measure generated by a random variable

X = X(ω)→ a random variable defined on a probability space< Ω,F , P >.

PX(A) = P (ω : X(ω) ∈ A) = P (X−1(A)), A ∈ B1.

Theorem 4.8. PX(A) is a probability measure defined on Borelσ-algebra B1.

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———————————(a) PX(A) ≥ 0;

(b) A1, A2, . . . ∈ B1, Ai ∩ Aj = ∅ ⇒ X−1(∪nAn) = ∪nX−1(An)and, therefore, PX(∪nAn) = P (X−1(∪nAn)) =

∑n P (X−1(An)) =∑

n PX(An);

(c) PX(R1) = P (X−1(R1)) = P (Ω) = 1.

———————————

Definition 4.6. The probability measure PX(A) is called a dis-tribution of the random variable X.

X = X(ω)→ a random variable defined on a probability space< Ω,F , P > and taking values in a space X with σ-algebra ofmeasurable sets BX .

PX(A) = P (ω : X(ω) ∈ A) = P (X−1(A)), A ∈ BX .

Theorem 4.9. PX(A) is a probability measure defined on Borelσ-algebra BX .

Definition 4.7. The probability measure PX(A) is called a dis-tribution of the random variable X.

2.2 Distribution function of a random variable

X = X(ω)→ a random variable defined on a probability space< Ω,F , P >;PX(A) = P (X(ω) ∈ A) = P (X−1(A)), A ∈ B1 → the distribu-tion of the random variable X.

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Definition 4.8 . The function FX(x) = PX((−∞, x]), x ∈ R1

is called the distribution function of a random variable X.

(1) The distribution PX(A) uniquely determines the distri-bution function FX(x) and, as follows from the continuationtheorem, the distribution function of random variable uniquelydetermines the distribution PX(A).

2.3 Properties of distribution functions

A distribution function FX(x) of a random variable X pos-sesses the following properties:

(1) FX(x) is non-decreasing function in x ∈ R1;

(2) FX(−∞) = limx→−∞ FX(x) = 0, FX(∞) = limx→∞ FX(x)= 1;(3) FX(x) is continuous from the right function, i.e., FX(x) =

limy≥x,y→x FX(y), x ∈ R1.

——————————-(a) x′ ≤ x′′ ⇒ (−∞, x′] ⊆ (−∞, x′′]⇒ FX(x′) = PX((−∞, x′]) ≤FX(x′′) = PX((−∞, x′′]);(b) xn → −∞⇒ zn = maxk≥n xk, ↓ −∞ ⇒ FX(xn) ≤ FX(zn) =PX((−∞, zn])→ 0 since ∩n(−∞, zn] = ∅;(c) xn → ∞ ⇒ zn = mink≥n xk, ↑ ∞ ⇒ FX(xn) ≥ FX(zn) =PX((−∞, zn])→ 1 since ∪n(−∞, zn] = R1;

(d) xn ≥ x, xn → x ⇒ zn = maxk≥n xk, ↓ x ⇒ FX(xn) ≤FX(zn) = PX((−∞, zn]) → FX(x) since ∩n(−∞, zn] = (−∞, x];

——————————-

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(4) P (X ∈ (a, b]) = PX((a, b]) = FX(b)− FX(a), a ≤ b;

(5) P (X = a) = PX(a) = FX(a)− FX(a− 0), a ≤ b;

——————————-(e) PX(a) = limn→∞(FX(a)−FX(a− 1

n)) = FX(a)−FX(a−0)since ∩n(a− 1

n , a] = a.——————————-

(6) Any distribution function has not more that n jumps withvalues ≥ 1

n for every n = 1, 2, . . . and, therefore, the set of alljumps is at most countable.——————————-(f) Let a1 < · · · < aN be some points of jumps with values≥ 1

n . Then ∪Nn=1X = an ⊆ −∞ < X < ∞. Thus, N/n ≤∑Nn=1 P (X = an) = P (∪Nn=1X = an ≤ P (−∞ < X < ∞) = 1

and thus N ≤ n.——————————-

2.4 Random variables with a given distribution func-tion

One can call any function F (x) defined on R1 a distributionfunction if it possesses properties (1)– (3).

According the continuation theorem every distribution func-tion uniquely determines (generates) a probability measure P (A)on B1 which is connected with this distribution function by therelation

P ((a, b]) = F (b)− F (a), a < b.

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Theorem 4.10. For any distribution function F (x) there ex-ists a random variable X that has the distribution functionFX(x) ≡ F (x).

——————————-(a) Choose the probability space < Ω = R1,F = B1, P (A) >where P (A) is the probability measure which is generated bythe distribution function F (x).(b) Consider the random variable X(ω) = ω, ω ∈ R1. ThenP (ω : X(ω) = ω ≤ x) = P ((−∞, x]) = F (x), x ∈ R1.——————————-

Let F (x) is distribution function. One can define the ”in-verse” function

F−1(y) = inf(x : F (x) ≥ y), 0 ≤ y ≤ 1.

.Random variable Y has an uniform distribution if it has the

following distribution function

FY (x) =

0 if x < 0,x if x ∈ [0, 1],1 if x > 1.

Theorem 4.11*. For any distribution function F (x) the ran-dom variable X = F−1(Y ), where Y is a uniformly distributedrandom variable, has the distribution function F (x).——————————-(a) Consider here only the case where F (x) is a continuous

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strictly monotonic distribution function. In this case F−1(y) =inf(y : F (x) = y) is also a continuous strictly monotonic func-tion and F−1(F (x)) = x.(b) F (x) = P (Y ≤ F (x)) = P (F−1(Y ) ≤ F−1(F (x))= P (X ≤ x), x ∈ R1.——————————-

Example

Let F (x) = 1− e−ax, x ≥ 0 be an exponential distribution func-tion In this case F−1(y) = −1

a ln(1 − y) and random variableX = −1

a ln(1 − Y ) has the exponential distribution functionwith parameter a.

2.5 Distribution functions of transformed random vari-ables

X → random variable with a distribution function FX(x) andthe corresponding distribution PX(A);f(x) : R1 → R1 is a Borel function.Af(x) = y ∈ R1 : f(y) ≤ x, x ∈ R1.

Theorem 4.12. The distribution function of the transformedrandom variable Y = f(X) is given by the following formula,

FY (x) = P (f(X) ≤ x) = P (X ∈ Af(x)) = PX(Af(x)), x ∈ R1.

Examples

(1) Y = aX + b, a > 0;Af(x) = (−∞, x−ba ];

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FY (x) = FX(x−ba ).

(2) Y = eaX , a > 0;Af(x) = ∅ if x ≤ 0 or (−∞, 1a lnx] if x > 0;FY (x) = I(x > 0)FX(1a lnx).

(3) Y = X2;Af(x) = ∅ if x ≤ 0 or [−

√x,√x] if x > 0;

FY (x) = I(x > 0)(FX(√x)− FX(−

√x− 0)).

3 Types of distribution functions

3.1 Discrete distribution functions

Definition 4.9. A distribution function F (x) is discrete if thereexists a finite or countable set of points A = a1, a2, . . . suchthat

∑n(F (an)− F (an − 0)) = 1.

If X is a random variable with the distribution function F (x)then

P (X ∈ A) =∑

an∈AP (X = an) =

∑an∈A

(F (an)− F (an − 0)).

Examples

(a) Bernoulli distribution;(b) Discrete uniform distribution;(c) Binomial distribution;(d) Poisson distribution;(e) Geometric distribution;

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3.2 Absolutely continuous distribution functions.

Definition 4.10. A distribution function F (x) is absolutelycontinuous if it can be represented in the following form

F (x) =∫ x

−∞f(y)dy, x ∈ R1,

where (a) f(y) is a Borel non-negative function; (b)∫∞−∞ f(y)dy =

1; (c) Lebesgue integration is used in the formula above (if f(y)is a Riemann integrable function then the Lebesgue integrationcan be replaced by Riemann integration).

Examples

(a) Uniform distribution;(b) Exponential distribution;(c) Normal (Gaussian) distribution;(d) Gamma distribution distribution;(e) Guachy distribution;(f) Pareto distribution.

3.3 Singular distribution functions.

Definition 4.11. A distribution function F (x) is singular if itis a continuous function and its set of points of growth SF hasLebesgue measure m(SF ) = 0 (x is point of growth if F (x+ ε)−F (x− ε) > 0 for any ε > 0).

Example*

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Define a continuous distribution function F (x) such that F (x)= 0 for x < 0 and F (x) = 1 for x > 1, which set of of points ofgrowth SF is the Cantor set, in the following way:

(a) Define function F (x) at the Cantor set in the following way:(1) [0, 1] = [0, 13 ] ∪ [13 ,

23 ] ∪ [23 , 1]: F (x) = 1

2 , x ∈ [13 ,23 ];

(2) [0, 13 ] = [0, 19 ] ∪ [19 ,29 ] ∪ [29 ,

13 ]: F (x) = 1

4 , x ∈ [19 ,29 ];

(3) [23 , 1] = [23 ,79 ] ∪ [79 ,

89 ] ∪ [89 , 1]: F (x) = 3

4 , x ∈ [79 ,89 ];

.......

(b) Define a function F (x) as continuous function in points thatdo not belong to the listed above internal intervals.

(c) The sum of length of all internal intervals, where the func-tion F (x) take constant values is equal

1

3+ 2 · 1

9+ 4 · 1

27+ · · · = 1

3

∞∑k=0

(2

3)k =

1

3· 1

1− 23

= 1.

3.4 Decomposition representation for distribution func-tions

Theorem 4.13 (Lebesgue)**. Any distribution functionF (x) can be represented in the form F (x) = p1F1(x)+p2F2(x)+p3F3(x), x ∈ R1 where (a) F1(x) is a discrete distribution func-tion, (a) F2(x) is an absolutely continuous distribution func-tion, (c) F3(x) is singular distribution function, (d) p1, p2, p2 ≥0, p1 + p2 + p3 = 1.

4 Multivariate random variables (random vectors)

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4.1 Random variables with values in a measurablespace

Let X is an arbitrary space and B(X ) is a σ-algebra of mea-surable subsets of X .

Definition 4.12. A random variable X = X(ω) defined on aprobability space < Ω,F , P > and taking values in the spaceX (with a σ-algebra of measurable subsets B(X )) is a mea-surable function acting from Ω → X , i.e., such function thatω : X(ω) ∈ A ∈ F for any A ∈ B(X ).

Examples

(1) X = R1,B(X ) = B1. In this case, X is a real-valued randomvariable;

(2) X = 0, 1 × · · · × 0, 1 (the product is taken n times),B(X ) is a σ-algebra of all subsets of X . A random variableX = (X1, . . . , Xn) is a Bernoulli vector which components areBernoulli random variables.

(3) X ia a metric space, B(X ) is a Borel σ-algebra of subsets ofX (the minimal σ-algebra containing all balls); X is a randomvariable taking values in the metric space X .

4.2 Random vectors

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X = Rk,B = Bk and P is a probability measure defined onBk.

Definition 4.13. A multivariate random variable (random vec-tor) is a random variable X = (X1, . . . , Xn) defined on a prob-ability space < Ω,F , P > and taking values taking values inthe space X = Rk (with a σ-algebra of measurable subsetsB(X ) = Bk).

(1) Every component of a random vector is a real-valued ran-dom variable defined on the same probability space.

(2) If Xk = Xk(ω), k = 1, . . . , n are real-valued random vari-ables defined on some probability space, then X = (X1, . . . , Xn)is a random vector defined on the same probability space.

4.3 Multivariate distribution functions

Definition. A multivariate distribution function of a ran-dom variable (random vector) X = (X1, . . . , Xn) is a functionF (x1, . . . , xn) defined for x = (x1, . . . , xn) ∈ Rn by the followingrelation

FX1,...,Xn(x1, . . . , xn) = P (X1 ≤ x1, . . . , Xn ≤ xn).

The multivariate distribution function possesses the follow-ing properties:

(1) limxk→−∞ FX1,...,Xn(x1, . . . , xn) = 0;

(2) FX1,...,Xn(x1, . . . , xn) non-decrease in every argument;

(3) limxk→∞,k=1,...,n FX1,...,Xn(x1, . . . , xn) = 1;

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(4) the multivariate distribution functions of the random vectors(X1, . . . , Xn) and (X1, . . . , Xk−1, Xk+1, . . . , Xn) are connected bythe following relation limxk→∞ FX1,...,Xn

(x1, . . . , xn) =FX1,...,Xk−1,Xk+1,...Xn

(x1, . . . , xk−1, xk+1, . . . , xn);

(5) P (X1 ∈ (a1, b1], . . . , Xn ∈ (a1, b1]) = FX1,...,Xn(b1, . . . , bn) −∑

k FX1,...Xn(b1, . . . , bk−1, ak, bk, . . . , bn) + · · ·

+(−1)nFX1,...,Xn(a1, . . . , an) ≥ 0;

(6) FX1,...,Xn(x1, . . . , xn) is continuous from above functions that

is limyk↓xk,k=1,...,n FX1,...,Xn(y1, . . . , yn) = FX1,...,Xn

(x1, . . . , xn).

Example

Let X = (X1, X2) is a two-dimensional random vector. Then

P (X1 ∈ (a1, b1], X2 ∈ (a2, b2]) = FX1,X2(b1, b2)

−FX1,X2(b1, a2)− FX1,X2

(a1, b2) + FX1,X2(a1, a2).

Theorem 4.13. A multivariate distribution functionFX1,...,Xn

(x1, . . . , xn) of a random vector X = (X1, . . . , Xn) uni-quely determines a probability measure PX(A) on the Borel σ-algebra Bk by its values on the cubes PX((a1, b1]×· · ·× (an, bn])= P (X1 ∈ (a1, b1], . . . , Xn ∈ (a1, b1]).

5 Independent random variables

5.1 Independent random variables

Definition Two random variables X and Y with distribu-tion functions, respectively, FX(x) and FY (y) are independent

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if the two-dimensional distribution of the random vector (X, Y )satisfies the relation

FX,Y (x, y) = FX(x) · FY (y), x, y ∈ R1.

(1) If random variables X and Y are independent then P (X ∈A, Y ∈ B) = P (X ∈ A) · P (Y ∈ B) for any A,B ∈ B1.

5.2 Mutually independent random variables

Definition 4.14. Random variables Xt, t ∈ T with distributionfunctions FXt

(x) are mutially independent if for any t1, . . . , tn, ti 6=tj the multivariate distribution function FXt1

,...,Xtn(x1, . . . , xn) of

the random vector (Xt1, . . . , Xtn) satisfies the relation

FXt1,...,Xtn

(x1, . . . , xn) = FXt1(x1)× · · · × FXtn

(xn).

LN Problems

1. LetA is a random event for a probability space< Ω,F , P >

and I = IA(ω) is a indicator of event A. Prove that I is a ran-dom variable.

2. Let X1, X2, . . . be a sequence of random variables definedon a probability space < Ω,F , P >. Let also Z = maxn≥1Xn

and I is an indicator of event A = Z < ∞. Let Y = Z · Iwhere the product is counted as 0 if Z = ∞, I = 0. Prove thatY is a random variable.

3. Let F (x) is a distribution function af a random vari-able X. Prove that P (a ≤ X ≤ b) = F (b) − F (a − 0) and

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P (a < X < b) = F (b− 0)− F (a).

4. Let random variable X has a continuous strictly mono-tonic distribution function F (x). Prove that the random vari-able Y = F (X) is uniformly distributed in the interval [0, 1].

5. Let Ω = ω1, ω2 . . . → be a discrete sample space, F0 →the σ-algebra of all subsets of Ω, and P (A) is a probability mea-sure on F . Let also X is a random variable defined on thediscrete probability space < Ω,F0, P >. Can the random vari-able X be a continuous or a singular distribution function?

5. Let random variable X has a distribution function F (x).What distribution function have random variables Y = aX2 +bX + c?

6 Let X and Y are independent random variables uniformlydistributed in the interval [0, 1]. What is the two-variate distri-bution function of the random vector Z = (X, Y )?

7. Let X1, . . . , Xn be independent random variables withthe same distribution functionF (x). What are the distribu-tion functions of random variables Z+

n = max(X1, . . . , Xn) andZ−n = min(X1, . . . , Xn)?

8. Give the proof of Theorem 4.7.

9. Give the proof of Theorem 4.9.

10. Give the proof of Theorem 4.9 for the case of a general

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distribution function (with possible jumps) [see G]

11. Give the proof of related to the example given in Section3.3.

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