Handbook of mathematics for engineers and scienteists part 156 pot

The linear transformation Y = X–a σ reduces the normal distribution with parameters a, σ2 and cumulative distribution function F x to the standard normal distribution with parameters 0,1

3

Figure 20.9 Probability density (a) and cumulate distribution (b) functions of normal distribution for a =1 ,

σ= 1/4

20.2.4-3 Normal distribution

A random variable X has the normal distribution with parameters (a, σ2) (see Fig 20.9a)

if its probability density function has the form

p (x) = 1

√

*

–(x – a)

2

2σ2

+

, x(–∞, ∞). (20.2.4.5)

The cumulative distribution function (see Fig 20.9b) and the characteristic function

have the form

F (x) = √1

2πσ

 x

–∞exp

*

–(t – a)

2

2σ2

+

dt,

f (t) = exp

*

iat– σ

2t2

2

+

,

(20.2.4.6)

and the numerical characteristics are given by the formulas

E{X}= a, Var{X}= σ2, Mode{X}= Med{X}= a, γ1 =0, γ2 =0,

μ k=

0, k=2m–1, m =1,2,

(2k–1)!! σ2k, k=2m , m =1,2,

The linear transformation Y = X–a σ reduces the normal distribution with parameters

(a, σ2) and cumulative distribution function F (x) to the standard normal distribution with

parameters (0,1) and cumulative distribution function

Φ(x) = √1

2π

 x

–∞ e

–t2/ dt; (20.2.4.7) moreover,Φ(–x) =1–Φ(x).

Remark 1 The values of the cumulative distribution functionΦ(x) of the standard normal distribution

are computed by the function NORMSDIST(z) in EXCEL software; for example, for Φ(2 ), the function call NORMSDIST(2) returns the value 0 9972

Remark 2. The values of the cumulative distribution function F (x) of the normal distribution are

computed by the function pnorm(x,mu,sigma) in MATHCAD software; to computeΦ(x), one should use

pnorm(x,0,1) For example, Φ(2 ) = pnorm(2,0,1) = 0 9972

The probability that a random variable X normally distributed with parameters (m, σ2)

lies in the interval (a, b) is given by the formula

P (a < ξ < b) = P

a – m

σ < ξ – m

σ < b – m

σ



=Φb – m

σ



–Φa – m

σ



(20.2.4.8)

A normally distributed random variable takes values close to its expectation with large

probability; this is expressed by the sigma rule

P(|X – m| ≥kσ) =2[1–Φ(k)] =

0

.3173 for k =1,

0.0456 for k =2,

0.0027 for k =3 The three-sigma rule is most frequently used

The fundamental role of the normal distribution is due to the fact that, under mild assumptions, the distribution of a sum of random variables is asymptotically normal as the number of terms increases The corresponding conditions are given in the central limit theorem

20.2.4-4 Cauchy distribution

A random variable X obeys the Cauchy distribution with parameters (a, λ) (λ > 0) (see

Fig 20.10a) if

π [λ2+ (x – a)2], x(–∞, ∞). (20.2.4.9)

1

x

3

Figure 20.10 Probability density (a) and cumulate distribution (b) functions of Cauchy distribution for a =1 ,

λ= 4

The cumulative distribution function has the form (see Fig 20.10b)

F (x) = 1

π arctanx – a

λ + 1

The numerical characteristics of a random variable that has a Cauchy distribution do not exist in the usual sense The expectation exists only in the sense of the Cauchy principal value (see Paragraph 10.2.2-3) and is given by the formula

E{X}= lim

T →∞

λ π

 T

–T

x dx

λ2+ (x – a)2 = a.

20.2.4-5 Chi-square distribution

A random variable X = χ2(n) has the chi-square distribution with n degrees of freedom if its probability density function has the form (see Fig 20.11a)

p (x) =

2n/2Γ(α/2)x

n/2 – 1e–x/2 for x >0,

(20.2.4.11)

1

1 2

3

1

2

3 3

Figure 20.11 Probability density (a) and cumulate distribution (b) functions of chi-square distribution for

n= 1(curve 1), n =2(curve 2), and n =3 (curve 3).

The cumulative distribution function can be written as (see Fig 20.11b)

2n/2Γ(α/2)

 x

0 ξ

n/2 – 1e–ξ/2dξ, (20.2.4.12) whereΓ(x) is a Gamma function.

Remark. The values χ2(x, n) of the cumulative distribution function of the chi-square distribution with n

degrees of freedom can be obtained using the expression 1 – CHIDIST(x; deg freedom) in EXCEL software For example, for the chi-square distribution with 10degrees of freedom at the point x =2, one gets χ2( 2 , 10 ) =

1 – CHIDIST( 2 ; 10 ) = 0 0037 A similar result is obtained if we use the function pchisq(x,n) in MATHCAD

software: χ2( 2 , 10 ) = pchisq( 2 , 10 ) = 0 0037

Main property of the chi-square distribution For an arbitrary n, the sum

X =

n



k=1

X2

k,

of squares of independent random variables obeying the standard normal distribution has

the chi-square distribution with n degrees of freedom.

THEOREM ON DECOMPOSITION Suppose that the sumn

k=1X

2

kof squares of independent

standard normally distributed random variables is expressed as the sum of L quadratic forms y j (X1, , X n)of ranks n j , respectively The variables y1, , y Lare independent

and obey the chi-square distributions with n1, , n L degrees of freedom if and only if

n1+· · · + n L = n.

THEOREM ON ADDITION,OR STABILITY PROPERTY The sum of L independent random variables y1, , y L obeying the chi-square distributions with n1, , n L degrees of

freedom, respectively, has the chi-square distribution with n = n1 +· · · + n L degrees of freedom

The characteristic function has the form

f (t) = (1–2it)–n/2, and the numerical characteristics are given by the formulas

E{χ2(n)}= n, Var{χ2(n)}=2n, α k = n(n +2)⋅ .⋅[n +2(k –1)],

γ1=2 2

n, γ2 = 12

n , Mode{χ2(n)}= n –2 (n≥ 2)

Relationship with other distributions:

1 For n =1, formula (20.2.4.11) gives the probability density function of the square X2

of a random variable with the standard normal distribution

2 For n =2, formula (20.2.4.11) gives the exponential distribution with parameter λ = 12

3 As n → ∞, the random variable X = χ2(n) has an asymptotically normal distribution

with parameters (n,2n)

4 As n → ∞, the random variable 2χ2(n) has an asymptotically normal distribution

with parameters (√

2n–1,1)

For the quantiles (denoted by χ2γ or χ2γ (n)), one has the approximation formula

χ2

γ (n)≈ 1

2(

√

2n–1+ t γ)2 (n≥ 30),

where t γis the quantile of the standard normal distribution

For γ close to0or1, it is more expedient to use the approximation given by the formula

χ2

γ (n)≈n



9n

3

The quantiles χ2γ (n) are tabulated; they can also be computed in EXCEL, MATHCAD,

and other software

Remark. Tables often list χ2–γ (n) rather than χ2γ (n).

20.2.4-6 Student’s t-distribution.

A random variable X = t(n) has Student’s distribution (t-distribution) with n degrees of

freedom (n >0) if its probability density function has the form (see Fig 20.12a)

p (x) = Γ(n+1

2 )

√

nπΓ(n

2)



2

n

–n+21

, x(–∞, ∞). (20.2.4.13) whereΓ(x) is Gamma function.

1

Figure 20.12 Probability density (a) and cumulate distribution (b) functions of Student’s t-distribution for

n= 3

The cumulative distribution function has the form (see Fig 20.12b)

F (x) = Γ(n+1

2 )

√

nπΓ(n

2)

 x

–∞



2

n

–n+21

dξ (20.2.4.14) Remark. The values of Student’s distribution function t(n) with n degrees of freedom can be computed,

for example, by using the function pt(x,n) in MATHCAD software.

Main property of Student’s distribution If η and χ2(n) are independent random variables and η has the standard normal distribution, then the random variable

t (n) = η n

χ2(n)

has Student’s distribution with n degrees of freedom.

The numerical characteristics are given by the formulas

E{t (n)}=0 (n >1), Var{t (n)}=

n–2 for n >2,

0 for n≤ 2, Mode{t (n)}= Med{t (n)}=0 (n >1),

α2k–1=0, α2k= n

k Γ(n/2– k) Γ(k +1/2)

√

π Γ(n/2) (2k < n),

γ1=0, γ2= 3(n –2)

n–4 (n >4).

Relationship with other distributions:

1 For n =1, Student’s distribution coincides with the Cauchy distribution

2 As n → ∞, Student’s distribution is asymptotically normal with parameters (0,1)

The quantiles of Student’s distribution are denoted by t γ (n) and satisfy

t γ (n) = –t1 –γ (n), |t1 –γ (n)|= t1–γ/2(n).

The quantiles t γ (n) are tabulated; they can also be computed in EXCEL, MATHCAD,

and other software

Student’s distribution is used when testing the hypothesis about the mean of a normally distributed population with unknown variance

20.2.5 Multivariate Random Variables

20.2.5-1 Distribution of bivariate random variable

Suppose that random variables X1, , X nare defined on a probability space (Ω, F, P );

then one says that an n-dimensional random vector X = (X1, , X n ) or a system of random

variables is given The random variables X1, , X ncan be viewed as the coordinates of

points in an n-dimensional space.

The distribution function F (x1, x2) = F X1,X2(x1, x2) of a two-dimensional random vector (X1, X2), or the joint distribution function of the random variables X1 and X2, is defined

as the probability of the simultaneous occurrence (intersection) of the events (X1< x1) and

(X2< x2); i.e.,

F (x1, x2) = F X1,X2(x1, x2) = P (X1< x1, X2 < x2) (20.2.5.1)

Geometrically, F (x1, x2) can be interpreted as the probability that the random point (X1, X2)

lies in the lower left infinite quadrant with vertex (x1, x2) (see Fig 20.13)

Given the joint distribution of random variables X1and X2, one can find the distributions

of each of the random variables X1and X2, known as the marginal distributions:

F X1(x1) = P (X1< x1) = P (X1 < x1, X2< +∞) = F X1 ,X2(x1, +∞),

F X2(x2) = P (X2< x2) = P (X1 < +∞, X2 < x2) = F X1,X2(+∞, x2) (20.2.5.2)

( , )x x

x

x

( , )X X

1

1

2

1

2

2

Figure 20.13 Geometrically interpretation of the distribution function F X1,X2(x1, x2).

The marginal distributions do not completely characterize the two-dimensional random

variable (X1, X2); i.e., the joint distribution of the random variables X1 and X2 cannot in general be reconstructed from the marginal distributions

Properties of the joint distribution function of random variables X1and X2:

1 The function F (x1, x2) is a nondecreasing function of each of the arguments

2 F (x1, –∞) = F (–∞, x2 ) = F (– ∞, –∞) =0

3 F (+ ∞, +∞) =1

4 The probability that the random vector lies in a rectangle with sides parallel to the coordinate axes is

P (a1 ≤X1< b1, a2≤X2 < b2) = F (b1, b2) – F (b1, a2) – F (a1, b2) + F (a1, a2).

5 The function F (x1, x2) is left continuous in each of the arguments

20.2.5-2 Discrete bivariate random variables

A bivariate random variable (X1, X2) is said to be discrete if each of the random variables

X1and X2is discrete.

If the random variable X1 takes the values x11, , x1m and the random variable X2 takes the values x21, , x2n , then the random vector (X1, X2) can take only the pairs of

values (x1i , x2j ) (i =1, , m, j =1, , n) It is convenient to describe the distribution of

a bivariate discrete random variable using the distribution matrix shown in Fig 20.14

Figure 20.14 Distribution matrix.

The entries p ij = P (X1= x1i , X2= x2j) of the distribution matrix are the probabilities of

the simultaneous occurrence of the events (X1= x1i ) and (X2= x2j ); P X1,i = p i1+· · · + p in

is the probability that the random variable X1takes the value x1i ; P X2,j = p1j+· · · + p mj

is the probability that the random variable X2 takes the value x2j; the last column (resp.,

row) shows the distribution of the random variable X1(resp., X2)

The distribution function of a discrete bivariate random variable can be determined by the formula

F (x1, x2) =

x1 <x1

x2j<x2

p ij. (20.2.5.3)

20.2.5-3 Continuous bivariate random variables

A bivariate random variable (X1, X2) is said to be continuous if its joint distribution function

F (x1, x2) can be represented as

F (x1, x2) =

 x2

–∞

 x1

–∞ p (y1, y2) dy1dy2, (20.2.5.4)

where the joint probability function p(x1, x2) = p X1,X2(x1, x2) is piecewise continuous The joint probability function can be expressed via the joint distribution function as follows:

p X1 ,X2(x1, x2) = p(x1, x2) = F x1x2(x1, x2) (20.2.5.5) Formulas (20.2.5.4) and (20.2.5.5) establish a one-to-one correspondence (up to sets of probability zero) between the joint probability functions and the joint distribution functions

of continuous bivariate random variables The differential p(x1, x2) dx1dx2 is called a

probability element Up to higher-order infinitesimals, the probability element is equal

to the probability for the random variable (X1, X2) to lie in the infinitesimal rectangle

(x1, x1+Δx1)×(x2, x2+Δx2).

The probability density function of the two-dimensional random variable (X1, X2),

which is also called the joint probability function of the random variables X1 and X2,

determines the probability density functions of the random variables X1and X2, which are

called the marginal probability functions of the two-dimensional random variable (X1, X2),

by the formulas

p X1(x1) =

 +∞

–∞ p X1,X2(x1, x2) dx2, p X2(x2) =

 +∞

–∞ p X1,X2(x1, x2) dx1. (20.2.5.6)

In the general case, the joint probability function cannot be reconstructed from the marginal probability functions, and hence the latter do not completely characterize the

bivariate random variable (X1, X2)

Properties of the joint probability function of random variables X1and X2:

1 The function p(x1, x2) is nonnegative; i.e., p(x1, x2) ≥ 0

2

 +∞

–∞

 +∞

–∞ p (x1, x2) dx1dx2 =1

3 P (a1< X1< b1, a2< X2< b2) =

 b1

a1

dx1

 b2

a2

p (x1, x2) dx2=

 b2

a2

dx2

 b1

a1

p (x1, x2) dx1

4 The probability for a two-dimensional random variable (X1, X2) to lie in a domain

D ⊂ R2 is numerically equal to the volume of the curvilinear cylinder with base D

bounded above by the surface of the joint probability function:

P [(X1, X2)D] =



(x1 ,x2 D p X1,X2(x1, x2) dx1dx2.

of each of the random variables X1and X2,... data-page="5">

Main property of Student’s distribution If η and χ2(n) are independent random variables and η has the standard normal distribution, then the random variable

t... x2) of a two-dimensional random vector (X1, X2), or the joint distribution function of the random variables X1 and X2,