The document discusses conditional probability density functions and conditional expected values. It defines the conditional probability density function of a random variable X given another random variable Y=y as f(x|y). This allows updating knowledge about X based on information about Y. The conditional expected value of X given Y=y is defined as the integral of xf(x|y)dx. An example calculates the conditional PDFs and expected value for two random variables with a given joint PDF.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but "quantized". The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
In this paper, the L1 norm of continuous functions and corresponding continuous estimation of regression parameters are defined. The continuous L1 norm estimation problem of one and two parameters linear models in the continuous case is solved. We proceed to use the functional form and parameters of the probability distribution function of income to exactly determine the L1 norm approximation of the corresponding Lorenz curve of the statistical population under consideration.
Pullbacks and Pushouts in the Category of Graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but "quantized". The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
In this paper, the L1 norm of continuous functions and corresponding continuous estimation of regression parameters are defined. The continuous L1 norm estimation problem of one and two parameters linear models in the continuous case is solved. We proceed to use the functional form and parameters of the probability distribution function of income to exactly determine the L1 norm approximation of the corresponding Lorenz curve of the statistical population under consideration.
Pullbacks and Pushouts in the Category of Graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
1. 1
As we have seen in section 4 conditional probability density
functions are useful to update the information about an
event based on the knowledge about some other related
event (refer to example 4.7). In this section, we shall
analyze the situation where the related event happens to be a
random variable that is dependent on the one of interest.
From (4-11), recall that the distribution function of X given
an event B is
(11-1)
.
)
(
)
)
(
(
|
)
(
)
|
(
B
P
B
x
X
P
B
x
X
P
B
x
FX
PILLAI
11. Conditional Density Functions and
Conditional Expected Values
2. 2
Suppose, we let
Substituting (11-2) into (11-1), we get
where we have made use of (7-4). But using (3-28) and (7-7)
we can rewrite (11-3) as
To determine, the limiting case we can let
and in (11-4).
.
)
( 2
1 y
Y
y
B
(11-3)
(11-2)
,
)
(
)
(
)
,
(
)
,
(
)
)
(
(
)
(
,
)
(
)
|
(
1
2
1
2
2
1
2
1
2
1
y
F
y
F
y
x
F
y
x
F
y
Y
y
P
y
Y
y
x
X
P
y
Y
y
x
F
Y
Y
XY
XY
X
.
)
(
)
,
(
)
|
( 2
1
2
1
2
1
y
y
Y
x y
y
XY
X
dv
v
f
dudv
v
u
f
y
Y
y
x
F
(11-4)
),
|
( y
Y
x
FX y
y
1
y
y
y
2
PILLAI
3. 3
This gives
and hence in the limit
(To remind about the conditional nature on the left hand
side, we shall use the subscript X | Y (instead of X) there).
Thus
Differentiating (11-7) with respect to x using (8-7), we get
(11-5)
(11-6)
.
)
(
)
,
(
)
|
(
lim
)
|
(
0 y
f
du
y
u
f
y
y
Y
y
x
F
y
Y
x
F
Y
x
XY
X
y
X
(11-7)
(11-8)
y
y
f
y
du
y
u
f
dv
v
f
dudv
v
u
f
y
y
Y
y
x
F
Y
x
XY
y
y
y
Y
x y
y
y
XY
X
)
(
)
,
(
)
(
)
,
(
)
|
(
.
)
(
)
,
(
)
|
(
|
y
f
du
y
u
f
y
Y
x
F
Y
x
XY
Y
X
.
)
(
)
,
(
)
|
(
|
y
f
y
x
f
y
Y
x
f
Y
XY
Y
X
PILLAI
4. 4
It is easy to see that the left side of (11-8) represents a valid
probability density function. In fact
and
where we have made use of (7-14). From (11-9) - (11-10),
(11-8) indeed represents a valid p.d.f, and we shall refer to it
as the conditional p.d.f of the r.v X given Y = y. We may also
write
From (11-8) and (11-11), we have
(11-9)
|
( , )
( | ) 0
( )
XY
X Y
Y
f x y
f x Y y
f y
,
1
)
(
)
(
)
(
)
,
(
)
|
(
|
y
f
y
f
y
f
dx
y
x
f
dx
y
Y
x
f
Y
Y
Y
XY
Y
X
(11-10)
(11-11)
,
)
(
)
,
(
)
|
(
|
y
f
y
x
f
y
x
f
Y
XY
Y
X (11-12)
PILLAI
).
|
(
)
|
( |
| y
x
f
y
Y
x
f Y
X
Y
X
5. 5
and similarly
If the r.vs X and Y are independent, then
and (11-12) - (11-13) reduces to
implying that the conditional p.d.fs coincide with their
unconditional p.d.fs. This makes sense, since if X and Y are
independent r.vs, information about Y shouldn’t be of any
help in updating our knowledge about X.
In the case of discrete-type r.vs, (11-12) reduces to
)
(
)
(
)
,
( y
f
x
f
y
x
f Y
X
XY
(11-13)
(11-14)
(11-15)
.
)
(
)
,
(
)
|
(
|
x
f
y
x
f
x
y
f
X
XY
X
Y
),
(
)
|
(
),
(
)
|
( |
| y
f
x
y
f
x
f
y
x
f Y
X
Y
X
Y
X
.
)
(
)
,
(
|
j
j
i
j
i
y
Y
P
y
Y
x
X
P
y
Y
x
X
P
PILLAI
6. 6
Next we shall illustrate the method of obtaining conditional
p.d.fs through an example.
Example 11.1: Given
determine and
Solution: The joint p.d.f is given to be a constant in the
shaded region. This gives
Similarly
and
(11-16)
,
otherwise
,
0
,
1
0
,
)
,
(
y
x
k
y
x
fXY
.
2
1
2
)
,
(
1
0
1
0 0
k
k
dy
y
k
dy
dx
k
dxdy
y
x
f
y
XY
)
|
(
| y
x
f Y
X ).
|
(
| x
y
f X
Y
x
y
1
1
Fig. 11.1
,
1
0
),
1
(
)
,
(
)
(
1
x
x
k
dy
k
dy
y
x
f
x
f
x
XY
X
(11-17)
.
1
0
,
)
,
(
)
(
0
y
y
k
dx
k
dx
y
x
f
y
f
y
XY
Y
(11-18)
PILLAI
7. 7
From (11-16) - (11-18), we get
and
We can use (11-12) - (11-13) to derive an important result.
From there, we also have
or
But
and using (11-23) in (11-22), we get
(11-19)
(11-20)
(11-21)
,
1
0
,
1
)
(
)
,
(
)
|
(
|
y
x
y
y
f
y
x
f
y
x
f
Y
XY
Y
X
.
1
0
,
1
1
)
(
)
,
(
)
|
(
|
y
x
x
x
f
y
x
f
x
y
f
X
XY
X
Y
)
(
)
|
(
)
(
)
|
(
)
,
( |
| x
f
x
y
f
y
f
y
x
f
y
x
f X
X
Y
Y
Y
X
XY
.
)
(
)
(
)
|
(
)
|
( |
|
x
f
y
f
y
x
f
x
y
f
X
Y
Y
X
X
Y (11-22)
| )
(
)
|
(
)
,
(
)
( dy
y
f
y
x
f
dy
y
x
f
x
f Y
Y
X
XY
X (11-23)
PILLAI
8. 8
Equation (11-24) represents the p.d.f version of Bayes’
theorem. To appreciate the full significance of (11-24), one
need to look at communication problems where
observations can be used to update our knowledge about
unknown parameters. We shall illustrate this using a simple
example.
Example 11.2: An unknown random phase is uniformly
distributed in the interval and where
n Determine
Solution: Initially almost nothing about the r.v is known,
so that we assume its a-priori p.d.f to be uniform in the
interval
.
)
(
)
|
(
)
(
)
|
(
)
|
(
|
|
dy
y
f
y
x
f
y
f
y
x
f
x
y
f
Y
Y
X
Y
Y
X
YX (24)
),
2
,
0
( ,
n
r
).
,
0
( 2
N ).
|
( r
f
).
2
,
0
(
PILLAI
9. 9
In the equation we can think of n as the noise
contribution and r as the observation. It is reasonable to
assume that and n are independent. In that case
since it is given that is a constant, behaves
like n. Using (11-24), this gives the a-posteriori p.d.f of
given r to be (see Fig. 11.2 (b))
where
,
n
r
)
,
(
)
|
( 2
N
r
f
θ (11-25)
θ
,
2
0
,
)
(
2
1
)
(
)
|
(
)
(
)
|
(
)
|
(
2
2
2
2
2
2
2
/
)
(
2
0
2
/
)
(
2
/
)
(
2
0
r
r
r
e
r
d
e
e
d
f
r
f
f
r
f
r
f
.
2
)
( 2
0
2
/
)
( 2
2
d
e
r
r
(11-26)
n
r
PILLAI
10. 10
Notice that the knowledge about the observation r is
reflected in the a-posteriori p.d.f of in Fig. 11.2 (b). It is
no longer flat as the a-priori p.d.f in Fig. 11.2 (a), and it
shows higher probabilities in the neighborhood of .
r
)
|
(
| r
f r
(b) a-posteriori p.d.f of
r
Fig. 11.2
Conditional Mean:
We can use the conditional p.d.fs to define the conditional
mean. More generally, applying (6-13) to conditional p.d.fs
we get
)
(
f
(a) a-priori p.d.f of
2
1
2
PILLAI
11. 11
( ) | ( ) ( | ) .
X
E g X B g x f x B dx
(11-27)
and using a limiting argument as in (11-2) - (11-8), we get
to be the conditional mean of X given Y = y. Notice
that will be a function of y. Also
In a similar manner, the conditional variance of X given Y
= y is given by
we shall illustrate these calculations through an example.
|
| )
|
(
| dx
y
x
f
x
y
Y
X
E Y
X
Y
X
(11-28)
)
|
( y
Y
X
E
.
)
|
(
| |
|
dy
x
y
f
y
x
X
Y
E X
Y
X
Y
(11-29)
.
|
)
(
)
|
(
|
)
|
(
2
|
2
2
2
|
y
Y
X
E
y
Y
X
E
y
Y
X
E
Y
X
Var
Y
X
Y
X
(11-30)
PILLAI
12. 12
Example 11.3: Let
Determine and
Solution: As Fig. 11.3 shows,
in the shaded area, and zero elsewhere.
From there
and
This gives
and
.
otherwise
,
0
,
1
|
|
0
,
1
)
,
(
x
y
y
x
fXY (11-31)
)
|
( Y
X
E ).
|
( X
Y
E
,
1
0
,
2
)
,
(
)
(
x
x
dy
y
x
f
x
f
x
x
XY
X
1
| |
( ) 1 1 | |, | | 1,
Y
y
f y dx y y
,
1
|
|
0
,
|
|
1
1
)
(
)
,
(
)
|
(
|
x
y
y
y
f
y
x
f
y
x
f
Y
XY
Y
X
.
1
|
|
0
,
2
1
)
(
)
,
(
)
|
(
|
x
y
x
x
f
y
x
f
x
y
f
X
XY
X
Y
(11-32)
(11-33)
x
y
1
Fig. 11.3
1
)
,
(
y
x
fXY
PILLAI
13. 13
Hence
It is possible to obtain an interesting generalization of the
conditional mean formulas in (11-28) - (11-29). More
generally, (11-28) gives
But
.
1
|
|
,
2
|
|
1
|)
|
1
(
2
|
|
1
2
|)
|
1
(
1
|)
|
1
(
)
|
(
)
|
(
2
1
|
|
2
1
|
|
|
y
y
y
y
x
y
dx
y
x
dx
y
x
f
x
Y
X
E
y
y
Y
X
.
1
0
,
0
2
2
1
2
)
|
(
)
|
(
2
|
x
y
x
dy
x
y
dy
x
y
yf
X
Y
E
x
x
x
x
X
Y
(11-34)
(11-35)
|
( )|
( ) ( ) ( ) ( ) ( , )
( ) ( , ) ( ) ( | ) ( )
( ) | ( )
X XY
XY X Y Y
E g X Y y
Y
E g X g x f x dx g x f x y dydx
g x f x y dxdy g x f x y dx f y dy
E g X Y y f y dy E E
( ) | .
g X Y y
|
( ) | ( ) ( | ) .
X Y
E g X Y y g x f x y dx
(11-36)
(11-37)PILLAI
14. 14
Obviously, in the right side of (11-37), the inner
expectation is with respect to X and the outer expectation is
with respect to Y. Letting g( X ) = X in (11-37) we get the
interesting identity
where the inner expectation on the right side is with respect
to X and the outer one is with respect to Y. Similarly, we
have
Using (11-37) and (11-30), we also obtain
,
)
|
(
)
( y
Y
X
E
E
X
E
.
)
|
(
)
( x
X
Y
E
E
Y
E
(11-38)
(11-39)
.
)
|
(
)
( y
Y
X
Var
E
X
Var
(11-40)
PILLAI
15. 15
Conditional mean turns out to be an important concept in
estimation and prediction theory. For example given an
observation about a r.v X, what can we say about a related
r.v Y ? In other words what is the best predicted value of Y
given that X = x ? It turns out that if “best” is meant in the
sense of minimizing the mean square error between Y and
its estimate , then the conditional mean of Y given X = x,
i.e., is the best estimate for Y (see Lecture 16
for more on Mean Square Estimation).
We conclude this lecture with yet another application
of the conditional density formulation.
Example 11.4 : Poisson sum of Bernoulli random variables
Let represent independent, identically
distributed Bernoulli random variables with
Yˆ
)
|
( x
X
Y
E
3,
2,
1,
,
i
Xi
q
p
X
P
p
X
P i
i
1
)
0
(
,
)
1
(
16. 16
and N a Poisson random variable with parameter that is
independent of all . Consider the random variables
Show that Y and Z are independent Poisson random variables.
Solution : To determine the joint probability mass function
of Y and Z, consider
.
,
1
Y
N
Z
X
Y
N
i
i
(11-41)
i
X
PILLAI
n
m
i
i
N
i
i
n
m
N
P
m
X
P
n
m
N
P
n
m
N
m
X
P
n
m
N
P
n
m
N
m
Y
P
n
m
N
m
Y
P
n
Y
N
m
Y
P
n
Z
m
Y
P
1
1
)
(
)
(
)
(
)
(
)
(
)
(
)
,
(
)
,
(
)
,
(
(11-42)
17. 17
)
)
,
(
~
( of
t
independen
are
and
that
Note
1
N
s
X
p
n
m
B
X i
n
m
i
i
(11-43)
( )!
! ! ( )!
m n
m n
m n
p q e
m n m n
!
)
(
!
)
(
n
q
e
m
p
e
n
q
m
p
).
(
)
( n
Z
P
m
Y
P
PILLAI
Thus
and Y and Z are independent random variables.
Thus if a bird lays eggs that follow a Poisson random
variable with parameter , and if each egg survives
)
(
~
)
(
~ and
q
P
Z
p
P
Y (11-44)
18. 18
with probability p, then the number of chicks that survive
also forms a Poisson random variable with parameter .
p
PILLAI