Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Mathematics (Математика). Учебное пособие
.pdf
81
According to the formula and according to the table calculate:
21
i
p
[
]
X 0 1 2 p 0,0625
0,375
0,5625
2*214*207*195*183*17
=X
++++
=
= 18,857 (years).
21/396
If we talk about mathematical statistics. In this case it is necessary to
consider the notion of mathematical expectation.
The mathematical expectation of a random variable X is the number
of views
the process;
s
×=
pxMX
å
, xi – all the different possible specific outcomes of
ii
=
i
1
– the probability of their occurrence.
Definition. The mathematical expectation of discrete random variable
is the sum of the products of all possible values of the random variable
on the probability.
n
...)(
2211
=+++==
å
=
i
pxpxpxpxXMm
iinnx
1
.
From the point of view of probability we can say that the mathematical expectation is approximately equal to the average of the observed
values of a random variable.
Properties of mathematical expectation.
1) The mathematical expectation of a constant is the constant: M(C) = C
2) Constant multiplier can be taken for a sign of the mathematical ex-
pectation: M(Cx) = CM(x)
3) The mathematical expectation of the product of two independent
random variables is equal to the product of their mathematical expectations: M(XY) = M(X)M(Y)
This property is valid for any number of random variables.
4) The mathematical expectation of the sum of two random variables
is equal to the sum of the mathematical expectations of components:
M(X+Y) = M(X)+M(Y)
This property is also true for any number of random variables.
Definition. Dispersion (scattering, variance) of a discrete random
variable is called the mathematical expectation of the squared deviation
of a random variable from its mathematical expectation.
2
)()( XMXMXD -=
Example. The distribution law of a random variable is:
82
To find the mathematical expectation and dispersion of a random var-
=×+×+×=
[
]
[
]
[
]
[X-M(X)]
2
2,25 0,25 0,25 p 0,0625
0,375
0,5625
=×+×+×=
X
D
[
]
x
x
-
iable. The mathematical expectation of the random variable is equal to:
XM
5,15625,02375,010625,00)(
Possible values of the squared deviation:
1
2
3
2
2
2
2
25,2)5,10()(
=-=- XMx
2
2
=-=- XMx
25,0)5,11()(
=-=- XMx
25,0)5,12()(
then
The dispersion is equal to:
375,05625,025,0375,025,00625,025,2)(
However, in practice this method of calculating variance uncomfortable, as results when a large number of values of the random variable for
large calculations.
We will show another way.
Theorem. The dispersion is equal to the difference between the
mathematical expectation of the square of a random variable X and the
square of its mathematical expectation.
2
2
)()()( XMXMXD -=
Twin regression – coupling equation of two variables x and y, for
which it is important to calculate the regression coefficient –
=
.
22
xyyx
-
b
Reading questions
1. How many ways of calculating dispersion discussed in this lec-
ture?
2. Does the concept of the arithmetic mean of the values connect
with the concept of mathematical expectation?
3. What is the difference in the calculation between the simple
arithmetic mean and the arithmetic mean weighted?
83
GLOSSARY
t
A
t
A
t
A
1
2
...nx
xxx
æö
ç÷
ç÷
ç÷
ç÷
ç÷
ç÷
ç÷
ç÷
ç÷
èø
(
)
...nxxxx
Matrix – a matrix is a rectangular array of numbers.
Zero Matrix – The m×n zero matrix is written as 0=0m×n and
defined by [0]ij=0, for all 1≤i≤m, 1≤j≤n.
Transpose of a Matrix. Given an m×n matrix A, its transpose is the
n×m matrix
Symmetric Matrix. The matrix A is symmetric if A=
Column vector Thus the n × 1 matrix
given by [
]ij=[A]ji,1≤i≤n,1≤j≤m
=
is also called a column
.
vector.
Row vector The 1 × n matrix
=
12
is called a row vector.
Matrix Equality. The m×n matrices A and B are equal, written A=B
provided [A]ij=[B]ij for all 1≤i≤m, 1≤j≤n.
Matrix Addition. Given the m×n matrices A and B, define the sum
of A and B as an m×n matrix, written A+B, according to
[A+B]ij=[A]ij+[B]ij, 1≤i≤m,1≤j≤n
Matrix Scalar Multiplication. Given the m×n matrix A and the sca-
lar α∈C, the scalar multiple of A is an m×n matrix, written αA and de-
fined according to [αA]ij=α[A]ij1≤i≤m,1≤j≤n
SubMatrix. Suppose that A is an m×n matrix. Then the submatrix
M(i|j) is the (m−1)×(n−1) matrix obtained from A by removing row
i and column j.
Determinant of a Matrix. Suppose A is a square matrix. Then its determinant, det(A)= |A| , is an element of C defined recursively by:
1. If A is a 1×1 matrix, then det(A)=[A]11.
2. If A is a matrix of size n with n≥2,
Then det(A)=[A]11det(M(1|1))−[A]12det(M(1|2))+[A]13det(M(1|3))−
84
n+1
1
n
11
kk
MA
ì
í
î
[A]
1n
det(M(1|n))
n
-
)1(
=
k
1
=
zyxF
=
zyxФ
+
k
1
Ma
,
kk
11
1
k
+
(1)
-=
0),,(
.
0),,(
–[A]14 det(M(1|4))+⋯+(−1)
aaа
А=
æ
ç
ç
...
...
ç
ç
ç
è
...
21
ö
n
11211
÷
aaa
÷
n
22221
÷
............
÷
÷
aaa
ø
nnnn
det A = å
Equation of a line in space
Canonical equation of the straight line in space
xx
-
m
yy
-
=
n
zz
-
000
=
p
.
Equation of a straight line passing through the two points in
space
xx
-
1
=
xx
-
12
yy
-
1
=
yy
-
12
zz
-
1
zz
-
12
.
Definition a × b Let a and b be two vectors in R3. Then a × b is de-
fined by the following two rules.
1. |a × b| = |a| |b| sin θ where θ is the included angle.
2. a × b · a = 0, a × b · b = 0, and a, b, a × b forms a right hand system.
List of literature.
1. Calculus: Early Transcendentals, James Stewart Cengage Learn-
ing, 2010.
2. Introduction to Analysis Edward D. Gaughan, American Mathe-
matical Society, 2009.
3. Calculus. Gilbert Strang by Wellesley-Cambridge Press, 1991
4. A First Course in Linear Algebra, Robert A. Beezer, Tacoma,
Washington, http://buzzard.ups.edu/ , Version 3.20 (Created: 2014-0224T20:51:53-08:00) 2013
5. Principles of Mathematical Modeling By Clive Dym, Harvey
Mudd College, Claremont, California, U.S.A. 2004. (ISBN: 978-0-12226551-8)
6. Mathematics, Statistics & Mathematical Education Abstract Book
85
7. From the 5th Annual International Conference on Mathematics,
Statistics & Mathematical Education, 13-16 June 2011, Athens, Greece.
Edited by Gregory T. Papanikos
8. Mathematical modeling . Ckassroom Notes in Applied mathemat-
ics. Edited Murray S, Klamkin. Philadelphia 1987.
9. http://www.collegeopentextbooks.org/
10. http://ocw.mit.edu/
11. http://en.wikibooks.org/
86
СHAPTER 2
-
-
ø
è
Section 1. Linear algebra
and analytical geometry
Topic 1. Matrix. Matrix Operations
Introduction. This workshop is devoted to the discussion of the following concepts: matrix, matrix equality, matrix operations, transposes
and symmetric matrices.
æ
Example 1. Given matrix А =
find 2А + В.
642
æ
ç
2А =
ç
ç
è
Solve and check yourself using the answers:
1. If А=
Answer:
2. If А=
Answer:
3. If А=
æ
ç
ç
è
æ
ç
ç
è
æ
ç
ç
ö
÷
, 2А + В =
824
÷
÷
646
ø
543
ö
, а В=
÷
-- 1423
ö
, а В=
÷
÷
ø
÷
ø
1163
æ
ç
ç
è
420
ö
÷
÷
--
328
ø
- 610
æ
ç
ç
è
53
60
æ
ç
ç
è
1023
- 5756
ç
ç
ç
è
æ
ç
ç
ç
è
æ
ç
ç
-
è
ö
÷
÷
ø
24
ö
, then 4А-8В …..?
÷
÷
71
ø
ö
æ
÷
ç
÷
ç
, В=
è
1073
1699
1067
321
ö
÷
412
÷
÷
323
ø
ö
÷
.
÷
÷
ø
123
ö
, then 2А+В …..?
÷
÷
203
ø
; B =
--- 0111
æ
ç
ç
ç
è
3247
ö
÷
÷
, then 2А-4В …..?
ø
431
ö
÷
875
, need to
÷
÷
421
ø
87
----
-
-
öø÷
öø÷
(
)
1081222
ö
÷
÷
1018616
ø
Answer:
æ
ç
ç
è
-
4. Find the sum and difference of matrices
1302
æ
А =
ç
4131
è
Answer:
ö
and В =
÷
ø
5020
æ
ç
1171
è
2604
æ
,
ç
8262
è
4322
æ
ç
3040
-
è
ö
, and calculate the matrix 2А.
÷
ø
Topic 2. Matrix Multiplication
Introduction. This workshop is devoted to the discussion of the following concepts: a row vector, a column vector matrix multiplication,
Scalar Matrix Multiplication. For a full understanding of the topic you
have an idea about matrices.
301
æ
ç
ç
ç
Example 1. Given matrix А =
è
number a = 2. You need to find the transposed matrix and to calculate
АТВ+aС.
121
ö
÷
-
440
÷
; ATB =
÷
213
ø
2
ö
÷
÷
; АТВ+aС =
÷
ø
æ
ç
ç
ç
è
9
æ
ç
4
ç
ç
10
è
121
-
440
213
-
2
æ
ö
ç
÷
4
ç
÷
+
÷
ç
2
ø
è
AT =
aC =
æ
ç
ç
ç
è
-
æ
ç
4
ç
ç
2
è
Example 2. To find the product of matrices А =
1
ö
АВ =
æ
÷
ç
4
÷
ç
×
÷
ç
3
ø
è
æ
ç
142
=
ç
ç
è
ö
÷
142
÷
÷
- 241
ø
1
ö
æ
ö
÷
ç
÷
3
÷
ç
÷
×
=
÷
ç
÷
2
ø
è
ø
7
ö
÷
÷
÷
ø
=
ö
æ
÷
ç
8
÷
ç
÷
ç
12
ø
è
×××
114121
æ
ö
ç
÷
=
×××
144424
ç
÷
ç
÷
×××
134323
è
ø
, В =
æ
ç
ç
ç
è
.
1
ö
æ
÷
ç
3
÷
ç
÷
ç
2
ø
è
1
æ
ç
ç
ç
è
142
4168
3126
-
1
ö
æ
÷
ç
2
÷
ç
÷
ç
1
ø
è
, С =
ö
÷
4
÷
и В = (2 4 1).
÷
3
ø
ö
÷
.
÷
÷
ø
×+×+×
213211
×-×+×
243410
×+×+×
223113
and the
ö
æ
÷
ç
÷
ç
=
÷
ç
ø
è
10
9
ö
÷
4
÷
÷
ø
;
88
(
)
1
(
)
(
)
ø
è
-
ö
æ
÷
ç
142
4
÷
ВА =
Example 3. To find the product of matrices А= (1 2), В =
ç
×
= 2×1 + 4×4 + 1×3 = 2 + 16 + 3 = 21.
÷
ç
3
ø
è
АВ = (1 2)×
43
ö
æ
÷
ç
ç
è
43
ö
æ
÷
ç
ç
è
=
÷
65
ø
124103 ++
=
1613
.
÷
65
ø
Solve and check yourself using the answers:
æ
ç
If А=
ç
Answer:
æ
ç
ç
è
32
65
- 125
ö
÷
÷
, а В=
61
ö
÷
÷
ø
07
æ
ç
ç
è
ö
÷
, then ВА …..?
÷
25
ø
Topic 3. Determinant of a Matrix
Introduction. This workshop is devoted to the discussion of the following concepts: sub matrix, Determinant of a Matrix, the rules to calculate the determinant of Matrices of Size Two and Determinant of a 3×3
matrix.
Example 1. To calculate the determinants of the second order
4 – 5
D = = 4
3 – 6
1 2
D = = 1
2 4
Example 2. To calculate the determinant of the fourth order, expanding it by elements of the 1st row
1 –1 2 2
D = 1 –2 3 2
4 –1 5 15
6 –8 7 9
.
(– 6) – (– 5) . 3 = – 24 + 15 = – 9;
.
4 – 2 . 2 = 0.
89
Solution:
–2 3 2 1 3 2
.
D = 1
1+1
(–1)
–1 5 15 + (–1) . (–1)
1+2
4 5 15 +
–8 7 9 6 7 9
1 –2 2 1 –2 3
4 –1 15 + 2 .(–1)
1+4
4 –1 5
= (–90 – 360 – 14 +
+ 2 .(–1)
1+3
6 –8 9 6 –8 7
+ 80 + 210 + 27) + (45 + 270 + 56 – 60 –105 –108) + 2(–9 –180 – 64 +
+ 12 + 72 +120) – 2(–7 – 60 – 96 +18 + 40 + 56) = –147 + 98 + 2 (–49) –
– 2 (–49) = –49.
Let's calculate this determinant, pre-converting it so that the 1st line
consists of zeros except for the element а11 in order to go straight from
the determinant of the 4-th order one determinant of the 3rd order
To do this, fold the 1st column to the 2nd, then the 1st column should
be multiplied by (-2) and sequentially fold with the 3rd and 4th columns.
1 –1 2 2 1 0 0 0 –1 1 0
.
D = 1 –2 3 2 = 1 –1 1 0 = 1
(–1)
1+1
3 –3 7 =
4 –1 5 15 4 3 –3 7 –2 –5 –3
6 –8 7 9 6 –2 –5 –3
= – 9 – 14 + 0 – 0 – 35 + 9 = – 49.
We have got the same result.
Example 3. To calculate the determinant of the matrix
121
ö
÷
-
320
÷
÷
113
ø
-
32
×=- )2310()3310(2)3112(
11
30
2
13
-
20
×+×-
1
13
А =
æ
ç
ç
ç
è
121
1
320
113
= –5 + 18 + 6 = 19.
=×+×+×-×-×-×-=
90
25
-
D+D
ö
æ
÷
ç
ç
è
. To find
÷
31
ø
Example 4. Let the given matrix А =
21
ö
æ
÷
ç
ç
è
÷
43
ø
, В =
the determinant det (AB).
1–й method: detA=4–6=–2; detB=15–2=13; det(AB)=detA×detB=–
26.
2–й method: AB=
×+××+×
32211251
æ
ç
ç
è
ö
÷
=
÷
×+××+×
34231453
ø
87
æ
ç
ç
è
ö
÷
,
÷
1819
ø
det(AB)=7×18–8×19=126–152=–26.
4301
2112
.
1230
3412
112
4
×-
230
412
Example 5. To find the determinant
-
-
4301
2112
1230
-
= -1
3412
211-
123
= -1(6 – 4) – 1(9 – 1) + 2(12 – 2) = -2 – 8 + 20 = 10.
341
211 -
3
123
341
-
-
212
-
×+
130
312
212 -
130
=
312
112 -
230
=
412
120 --
130
= 2(0 – 2) – 1(0 – 6) = 2.
312
320 --
230
= 2(-4) – 3(-6) = -8 + 18 = 10.
412
The value of the determinant is….. -10 + 6 – 40 = -44.
Solve and check yourself using the answers:
32
If
=D
1
32
и
40
2
-
=D
40
, then
2
…..?
21
Answer: 24.
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]