Research Journal of Applied Sciences, Engineering and Technology 4(2): 86-89, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: September 23, 2011
Accepted: November 04 , 2011
Published: January 15, 2012
Application of Mathematical Expectation in Economical Field
Wen Jing-Hua, Zhang Mei and Yu Xi
School of Information, Guizhou Financial Institute, Guiyang, 550004, China
Abstract: In the economical field, mathematical expectation is one of the important digital characters of the
random variables. Many decision variables in economy problem are random variables; it is difficult to define
their concrete distribution. In this study, based on mathematical statistics method and finite difference theory,
we resolve the economical decision questions using the mathematical expectation. Moreover, the application
of mathematical expectation in economical question was analyzed by example from two aspects of investment
decision and sales profits. The results in our analysis show that mathematical expectation plays a great
application in production and sales profit.
Key words: Economic question, investment decision, mathematical expectation, sales profits
mathematical expectation of a DRVFP can be translated
into solving mathematical expectation of a series of RVIP.
Using the method of mathematical expectation, Gao and
Su (2009) study the fixed point theorem of Banach in the
probabilistic metric spaces and to give a probabilistic
version of Banach contraction map- ping principle. In
breif, mathematical expectation is an important digital
character which reflects mean level of total value of
random variable, and it widely used in the work of
economic management. For that reason, this paper
researches the application of mathematical expectation in
the field of economics from the point of view of quantum
theory in mathematics, and taking mathematical
expectation as he key point, digging its own fascination of
mathematical expectation breadth wise and length wise,
in order to provide gist to make decision reasonably for
economic management and actual life.
INTRODUCTION
In society science, the first application field of
mathematic is undoubtedly economics. Mark thought that
the flag to balance a subject mature or not is viewing its
application degree of mathematics. The economics
developed rapidly in the 20th century, and the application
of its mathematical tools and model are more and more
wide and deep, this is the progress no doubt. So it
becomes a trend that researching the application of
mathematic in the field of economics (Karolina, 2007;
Ashimov et al., 2010).
Quantum theory is a subject researching statistical
regularity of random phenomenon by the aspect of
quantity, and the distribution function of random variable
can reflect statistical regularity of random variable
roundly. But in so many economical management or
decision work, on the one hand for it is not an easy thing
to obtain the distribution function of random variable,
moreover it does not need to describe random variable
roundly for some practical problem, and it only need to
know some important digital character which can reflect
random variable. Gusev (2011) deals with finding ways of
reducing the variance of a mathematical expectation
estimate for the functional of a diffusion process moving
in a domain with an absorbing boundary. The estimate of
mathematical expectation of the functional is obtained
based on a numerical solution of Stochastic Differential
Equations (SDEs) by using the Euler method. The
character and an algorithm about DRVIP (Discrete
Random Variable with Interval Probability) and the
second kind DRVFP (discrete random variable with crisp
event-fuzzy probability) are researched in (Xiao and Lu,
2005). Using the fuzzy resolution theorem, the solving
MATHEMATICAL EXPECTATIONS
Mathematical expectation of discrete random
variable: Supposing the probability distribution of
discrete random variable X is P{X = xk} = pk,k = 1, 2, If

progression
 xk pk
k 1
is absolute convergence, then the
sum of the progression is called Mathematical expectation
of random variable X, it is called expectation or mean
value for short, denoted asE(X) (Li and Xin-quan, 2008),
namely:

E ( X )   xk pk
k 1
Corresponding Author: Wen Jing-Hua, School of Information, Guizhou Financial Institute, Guiyan, 550004, China
86
(1)
Res. J. Appl Sci. Eng. Technol., 4(2): 86-89, 2012
Toward discrete random variable X, E(X), is the sum of
product of each possible value of X and its corresponding
probability? In the condition of not predicting confusion,
it can be marked as Ex.
Attention: For X is random variable, there is no
special conditions of its dereferencing order. Requiring
function Z = g(x, y) of the random variable X,Y. (g is a
continuous function), Z is one-dimensional random
variable. If the joint distribution law of two dimension
discrete random variable (X, Y) is P{X = xi, y = yj} = pij,
i, j = 1, 2..., and

progression
 xk pk
k 1

j 1 i 1
(2)
If the probability density of two dimension
continuous random variable (x, y) is f(x, y), and it is
absolute convergence. Then the Mathematical
expectation of Z = g(x, y) is:

E ( X )   xf ( x )dx
E ( Z )  E g ( X , Y )
is called as Mathematical expectation of X (Yao, 2001). It
means that the Mathematical expectation of continuous
random variable X is the integral of product of value X
and probability density f(x) in the infinite interval(-4,+4).


     g ( x , y ) f ( x , y )dxdy
(3)
Properties of mathematical expectation:
Mathematical expectation of continuous random
function: Suppose x as the discrete random variable,
probability distribution is P{X = xk} = pk, k = 1, 2 ,….
If progression:
C
C

 g ( x k ) pk
C
k 1
C
is absolute convergence. Y = g(x) Is the function of
random variable X .Its Mathematical expectation is:

 g ( x k ) pk
Mathematical expectation of constant C is itself.
Namely, EC) = C
Mathematical expectation of Linear function of
random variable equals the same linear function of
this mathematical expectation of random variable.
Namely:
E(kx+b) = E(kx)+b = kE(x)+b
The mathematical expectation of the algebraic sum of
two random variables equal the algebraic sum of
mathematical expectation of those two random
variables.
Namely: E(X±Y) = E(X)±E(Y)
k 1
This property can be extended to any case of finite
number of random.
Suppose X as the continuous random variable; probability

density is f(x).If infinite integral
 g( x ) f ( x )dx
is
Namely: E(X1±X2±..n) = E(X1)±E(X2)±...E(Xn)
absolute convergence (HEU, 2003). Y = g(x) Is the
function of random variable X. Its Mathematical
expectation is:
C


ij
is
absolute convergence, and
 g( x) f ( x)dx
j


E (Y )  E[ g ( X )] 
i 1
E ( Z )  E  g ( X , Y )    g ( xi , y j ) pi j
0
E (Y )  E g ( X ) 
j 1
is absolute convergence, the Mathematical expectation
(Zhang, 2004) of Z = g(x, y) is:
Mathematical expectation of continuous random
variable: Suppose continuous random variable X has
 xf ( x)dx

i
to guarantee the sum of the progression (Sheng et al.,
2009), and it is independent of order of each item.
probability density f(x) . If integral

  g( x , y ) p
absolute convergence is in order
.
Mathematical Expectation of Two-Dimension
Random Variable Function Z = g(x, y): Suppose Z is the
In particular, arithmetic mean of n random variables
also is a random variable. Its mathematical
expectation equal arithmetic mean of mathematical
expectation of n random variable.
Namely:
87
1 n
1 n
E (  Xi )   E ( Xi )
n i 1
n i 1
(4)
Res. J. Appl Sci. Eng. Technol., 4(2): 86-89, 2012
Table 1: The table of a variety of investment annual yield distribution
Excellent
Good
Bad
Building property x
11
3
-3
Landed estate y
6
4
-1
Business Z
10
2
-2
C
Application of mathematical expectation in sales
profit: The weekly sales of air-condition . in store is a
random variable. Its distribution law is:
P(  k ) 1 20 , k  11,12,13,..., 30.
Mathematical expectation of product of two mutual
independence random variables equals their product
of their mathematical expectations (chen, 2005; Gou
and 2 hang, 2005).
Every week, the purchasing amount is a Integer
among[11, 30].Each sale of an air conditioner can profit 500
Yuan. If supply exceeds demand, each extra air
conditioner is required to pay 100 Yuan safekeeping fee.
If demand exceeds supply, we can supply from other
stores. Now each air conditioner just profits 200 Yuan.
How much the initial amount do we purchase per week
(including the remainder about last week), then we can
get the biggest profits?
Suppose the initial purchasing amount is x per week.
Weekly profit is a random variable n, then:
Namely: E(XY) = E(X).E(Y). This property can be
extended to the case of finite number of independent
random.
Namely: E(X1X2 ..., xn) = E(X1).E(X2)... E(Xn)
APPLICATION OF MATHEMATICAL
EXPECTATION IN ECONOMICAL FIELD
Application of mathematical expectation in investment
decisions: Somebody has one fund .It can be put into
three projects: building] property X, landed estate Y,
business Z. Its revenue has related with market status. If
the future market is divided into three levels as excellent,
good and bad. The probability of its occurrence were
P1 = 0.2, P2 = 0.7, P3 = 0.1.According to market research
situation, we can get the variety of investment income in
different levels (10 thousand Yuan), following the
Table 1.
 500  100( x   )  600  100 x ,   1112
, ,..., x  1,

   500 ,   x ,
 500 x  200(  x )  300 x  200 ,   x  1, x  2,..., 30.

P(  k ) 
 E 
C
C
C

E(X) = 11×0.2+3×0.7+(!3)×0.1 = 4.0
E(y) = 6×0.2+4×0.7+(!1)×0.1 = 3.9
E(Z) = 6×0.2+2×0.7+(!2)×0.1 = 3.2
C
C
C
x 1
30
 11
  x 1
 30  5x   25x   15x  10 
In order to get the Max of E  , set d(E  )/dx = 0, then -
D(X) = (11!4) ×0.2+(3!4) ×0.7+(!3!4) ×0.1 =
15.40
D(Y) = (6!3.9)20.2+(4!3.9)2×0.7 +(!1!3.9)2×0.1 =
3.29
D(Z) = (10-3.2)2×0.2+(2!3.2)2×0.7(!2!3.2)2×0.1 =
12.96
2
1
1 30
 500 x 
 (300 x  200 )
20
20   x 1
11   x  1
 30
 x  11  5x x  11  25x
2
 x  1  30
 15x30  x   10
30  x 
2
  10 x 2  510 x  3000
Based on the mathematical expectation, invested in
the building] property, he can get the largest average
income. He may choose real estate, but at the same time
he should also consider the risks of investing. Then their
variance was observed:
2
1 x 1
 (600  100 x )
20  11

Question: How to invest in the most reasonable?
Consider the mathematical expectation:
1
, k  11,12,13,..., 30.
20
2
20x+510 = 0 and x = 25.5.
Because x is the positive integer, x = 25 or x = 26,
that means the initial purchasing amount per week is 25
or 26. Then we can get the max profits.
From the analysis above, mathematical expectation
plays a great application in production and sales profit.
The greater the variance, the fluctuations are larger in
income. Therefore, the risk is greater. According to
variance, the risk in building property is greater than that
of landed estate. Taking income and risk into
consideration, it is better to invest in landed estate.
Although the average income has reduced 1000 yuan, the
risk has reduced more than 50%.
ACKNOWLEDGMENT
Thank Project Supported by Basic Research Fund of
Guizhou Provincial Science and Technology Department
(QianKeHeJZi[2011]2191). Thank Project Supported by
Basic Research Fund of Guizhou Provincial Science and
Technology Department (J[2010]2024). Thank Project
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Res. J. Appl Sci. Eng. Technol., 4(2): 86-89, 2012
Supported by Nomarch Fund of Guizhou Provincial
excellence science and technology education person with
ability ((2007)41).
HEU, Applied Mathematics Department in Harbin
Engineering University, 2003. Probability Theory
and Mathematical Statistics. Harbin Engineering
University Press, Harbin.
Karolina, K., 2007. A mathematical economic model of
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Li, B.N. and Z. Xin-Quan, 2008. Probability Theory and
Mathematical Statistics. Higher Education Press,
Beijing.
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