Fuzzy random variables and Kolomogrov’s important resultsinventionjournals
:In this paper an attempt is made to transform Kolomogrov Maximal inequality, Koronecker Lemma, Loeve’s Lemma and Kolomogrov’s strong law of large numbers for independent, identically distributive fuzzy Random variables. The applications of this results is extensive and could produce intensive insights on Fuzzy Random variables
Fuzzy random variables and Kolomogrov’s important resultsinventionjournals
:In this paper an attempt is made to transform Kolomogrov Maximal inequality, Koronecker Lemma, Loeve’s Lemma and Kolomogrov’s strong law of large numbers for independent, identically distributive fuzzy Random variables. The applications of this results is extensive and could produce intensive insights on Fuzzy Random variables
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
Comparative structure of adrenal gland in vertebrates
Random variable, distributive function lect3a.ppt
1. 1
3. Random Variables
Let (, F, P) be a probability model for an experiment,
and X a function that maps every to a unique
point the set of real numbers. Since the outcome
is not certain, so is the value Thus if B is some
subset of R, we may want to determine the probability of
“ ”. To determine this probability, we can look at
the set that contains all that maps
into B under the function X.
,
,
R
x
.
)
( x
X
B
X
)
(
)
(
1
B
X
A
R
)
(
X
x
A
B
Fig. 3.1 PILLAI
2. 2
Obviously, if the set also belongs to the
associated field F, then it is an event and the probability of
A is well defined; in that case we can say
However, may not always belong to F for all B, thus
creating difficulties. The notion of random variable (r.v)
makes sure that the inverse mapping always results in an
event so that we are able to determine the probability for
any
Random Variable (r.v): A finite single valued function
that maps the set of all experimental outcomes into the
set of real numbers R is said to be a r.v, if the set
is an event for every x in R.
)
(
1
B
X
A
)).
(
(
"
)
(
"
event
the
of
y
Probabilit 1
B
X
P
B
X
(3-1)
)
(
1
B
X
.
R
B
)
(
X
)
(
| x
X
)
( F
PILLAI
3. 3
Alternatively X is said to be a r.v, if where B
represents semi-definite intervals of the form
and all other sets that can be constructed from these sets by
performing the set operations of union, intersection and
negation any number of times. The Borel collection B of
such subsets of R is the smallest -field of subsets of R that
includes all semi-infinite intervals of the above form. Thus
if X is a r.v, then
is an event for every x. What about
Are they also events ? In fact with since
and are events, is an event and
hence is also an event.
}
{ a
x
a
b
?
, a
X
b
X
a
b
X
}
{ b
X
a
b
X
a
X
a
X
a
X
c
)
(
| x
X
x
X
F
B
X
)
(
1
}
{ a
X
(3-2)
PILLAI
4. 4
Thus, is an event for every n.
Consequently
is also an event. All events have well defined probability.
Thus the probability of the event must
depend on x. Denote
The role of the subscript X in (3-4) is only to identify the
actual r.v. is said to the Probability Distribution
Function (PDF) associated with the r.v X.
1
a
X
n
a
1
}
{
1
n
a
X
a
X
n
a
)
(
| x
X
.
0
)
(
)
(
|
x
F
x
X
P X
(3-4)
)
(x
FX
(3-3)
PILLAI
5. 5
Distribution Function: Note that a distribution function
g(x) is nondecreasing, right-continuous and satisfies
i.e., if g(x) is a distribution function, then
(i)
(ii) if then
and
(iii) for all x.
We need to show that defined in (3-4) satisfies all
properties in (3-6). In fact, for any r.v X,
,
0
)
(
,
1
)
(
g
g
,
0
)
(
,
1
)
(
g
g
,
2
1 x
x ),
(
)
( 2
1 x
g
x
g
),
(
)
( x
g
x
g
(3-6)
)
(x
FX
(3-5)
PILLAI
6. 6
1
)
(
)
(
|
)
(
P
X
P
FX
.
0
)
(
)
(
|
)
(
P
X
P
FX
(i)
and
(ii) If then the subset
Consequently the event
since implies As a result
implying that the probability distribution function is
nonnegative and monotone nondecreasing.
(iii) Let and consider the event
since
,
2
1 x
x ).
,
(
)
,
( 2
1 x
x
,
)
(
|
)
(
| 2
1 x
X
x
X
1
)
( x
X
.
)
( 2
x
X
),
(
)
(
)
(
)
( 2
2
1
1 x
F
x
X
P
x
X
P
x
F X
X
(3-9)
(3-7)
(3-8)
,
1
2
1 x
x
x
x
x n
n
.
)
(
| k
k x
X
x
A
(3-10)
,
)
(
)
(
)
( k
k x
X
x
X
x
X
x
(3-11)
PILLAI
7. 7
using mutually exclusive property of events we get
But and hence
Thus
But the right limit of x, and hence
i.e., is right-continuous, justifying all properties of a
distribution function.
).
(
)
(
)
(
)
( x
F
x
F
x
X
x
P
A
P X
k
X
k
k
(3-12)
,
1
1
k
k
k A
A
A
.
0
)
(
lim
hence
and
lim
1
k
k
k
k
k
k
A
P
A
A
(3-13)
.
0
)
(
)
(
lim
)
(
lim
x
F
x
F
A
P X
k
X
k
k
k
),
(
)
( x
F
x
F X
X
)
(x
FX
(3-14)
,
lim
x
xk
k
PILLAI
8. 8
Additional Properties of a PDF
(iv) If for some then
This follows, since implies
is the null set, and for any will be a subset
of the null set.
(v)
We have and since the two events
are mutually exclusive, (16) follows.
(vi)
The events and are mutually
exclusive and their union represents the event
0
)
( 0
x
FX ,
0
x .
,
0
)
( 0
x
x
x
FX
(3-15)
0
)
(
)
( 0
0
x
X
P
x
FX
0
)
( x
X
)
(
,
0 x
X
x
x
).
(
1
)
( x
F
x
X
P X
(3-16)
,
)
(
)
(
x
X
x
X
.
),
(
)
(
)
( 1
2
1
2
2
1 x
x
x
F
x
F
x
X
x
P X
X
(3-17)
}
)
(
{ 2
1 x
X
x
)
( 1
x
X
.
)
( 2
x
X
PILLAI
9. 9
(vii)
Let and From (3-17)
or
According to (3-14), the limit of as
from the right always exists and equals However the
left limit value need not equal Thus
need not be continuous from the left. At a discontinuity
point of the distribution, the left and right limits are
different, and from (3-20)
).
(
)
(
)
(
x
F
x
F
x
X
P X
X
(3-18)
,
0
,
1
x
x .
2 x
x
),
(
lim
)
(
)
(
lim
0
0
x
F
x
F
x
X
x
P X
X (3-19)
).
(
)
(
)
(
x
F
x
F
x
X
P X
X
(3-20)
),
( 0
x
FX )
(x
FX 0
x
x
).
( 0
x
FX
)
( 0
x
FX ).
( 0
x
FX )
(x
FX
.
0
)
(
)
(
)
( 0
0
0
x
F
x
F
x
X
P X
X
(3-21)
PILLAI
10. 10
Thus the only discontinuities of a distribution function
are of the jump type, and occur at points where (3-21) is
satisfied. These points can always be enumerated as a
sequence, and moreover they are at most countable in
number. Example
3.1: X is a r.v such that Find Solution: For
so that and for so that
(Fig.3.2)
Example 3.2: Toss a coin. Suppose the r.v X is
such that Find
)
(x
FX
0
x
.
,
)
(
c
X ).
(x
FX
,
)
(
,
x
X
c
x ,
0
)
(
x
FX
.
,T
H
.
1
)
(
,
0
)
(
H
X
T
X
)
(x
FX
x
c
1
Fig. 3.2
).
(x
FX
.
1
)
(
x
FX
,
)
(
,
x
X
c
x
PILLAI
11. 11
Solution: For so that
•X is said to be a continuous-type r.v if its distribution
function is continuous. In that case for
all x, and from (3-21) we get
•If is constant except for a finite number of jump
discontinuities(piece-wise constant; step-type), then X is
said to be a discrete-type r.v. If is such a discontinuity
point, then from (3-21)
,
)
(
,
0
x
X
x .
0
)
(
x
FX
3.3)
(Fig.
.
1
)
(
that
so
,
,
)
(
,
1
,
1
)
(
that
so
,
)
(
,
1
0
x
F
T
H
x
X
x
p
T
P
x
F
T
x
X
x
X
X
).
(
)
(
i
X
i
X
i
i x
F
x
F
x
X
P
p (3-22)
)
(x
FX
)
(
)
( x
F
x
F X
X
.
0
x
X
P
)
(x
FX
)
(x
FX
x
Fig.3.3
1
q
1
i
x
PILLAI
12. 12
From Fig.3.2, at a point of discontinuity we get
and from Fig.3.3,
Example:3.3 A fair coin is tossed twice, and let the r.v X
represent the number of heads. Find
Solution: In this case and
.
1
0
1
)
(
)
(
c
F
c
F
c
X
P X
X
.
0
)
0
(
)
0
(
0 q
q
F
F
X
P X
X
,
,
,
, TT
TH
HT
HH
).
(x
FX
.
0
)
(
,
1
)
(
,
1
)
(
,
2
)
(
TT
X
TH
X
HT
X
HH
X
3.4)
(Fig.
.
1
)
(
)
(
,
2
,
4
3
,
,
)
(
,
,
)
(
,
2
1
,
4
1
)
(
)
(
)
(
)
(
,
1
0
,
0
)
(
)
(
,
0
x
F
x
X
x
TH
HT
TT
P
x
F
TH
HT
TT
x
X
x
T
P
T
P
TT
P
x
F
TT
x
X
x
x
F
x
X
x
X
X
X
X
PILLAI
13. 13
From Fig.3.4,
Probability density function (p.d.f)
The derivative of the distribution function is called
the probability density function of the r.v X. Thus
Since
from the monotone-nondecreasing nature of
.
2
/
1
4
/
1
4
/
3
)
1
(
)
1
(
1
X
X F
F
X
P
)
(x
FX
)
(x
fX
.
)
(
)
(
dx
x
dF
x
f X
X
,
0
)
(
)
(
lim
)
(
0
x
x
F
x
x
F
dx
x
dF X
X
x
X
(3-23)
(3-24)
),
(x
FX
)
(x
FX
x
Fig. 3.4
1
4
/
1
1
4
/
3
2
PILLAI
14. 14
it follows that for all x. will be a
continuous function, if X is a continuous type r.v.
However, if X is a discrete type r.v as in (3-22), then its
p.d.f has the general form (Fig. 3.5)
where represent the jump-discontinuity points in
As Fig. 3.5 shows represents a collection of positive
discrete masses, and it is known as the probability mass
function (p.m.f ) in the discrete case. From (3-23), we
also obtain by integration
Since (3-26) yields
0
)
(
x
fX )
(x
fX
,
)
(
)
(
i
i
i
X x
x
p
x
f
).
(x
FX
(3-25)
.
)
(
)
( du
u
f
x
F
x
x
X
(3-26)
,
1
)
(
X
F
,
1
)
(
dx
x
fx
(3-27)
i
x
)
(x
fX
x
i
x
i
p
Fig. 3.5
)
(x
fX
PILLAI
15. 15
which justifies its name as the density function. Further,
from (3-26), we also get (Fig. 3.6b)
Thus the area under in the interval represents
the probability in (3-28).
Often, r.vs are referred by their specific density functions -
both in the continuous and discrete cases - and in what
follows we shall list a number of them in each category.
.
)
(
)
(
)
(
)
(
2
1
1
2
2
1 dx
x
f
x
F
x
F
x
X
x
P
x
x
X
X
X
(3-28)
Fig. 3.6
)
(x
fX )
,
( 2
1 x
x
)
(x
fX
(b)
x
1
x 2
x
)
(x
FX
x
1
(a)
1
x 2
x
PILLAI
16. 16
Continuous-type random variables
1. Normal (Gaussian): X is said to be normal or Gaussian
r.v, if
This is a bell shaped curve, symmetric around the
parameter and its distribution function is given by
where is often tabulated. Since
depends on two parameters and the notation
will be used to represent (3-29).
.
2
1
)
(
2
2
2
/
)
(
2
x
X e
x
f (3-29)
,
,
2
1
)
(
2
2
2
/
)
(
2
x
y
X
x
G
dy
e
x
F
(3-30)
dy
e
x
G y
x
2
/
2
2
1
)
(
)
,
( 2
N
X
)
(x
fX
x
Fig. 3.7
)
(x
fX
,
2
PILLAI
17. 17
2. Uniform: if (Fig. 3.8)
,
),
,
( b
a
b
a
U
X
otherwise.
0,
,
,
1
)
(
b
x
a
a
b
x
fX
(3.31)
)
(x
fX
x
a b
a
b
1
Fig. 3.8
3. Exponential: if (Fig. 3.9)
)
(
X
otherwise.
0,
,
0
,
1
)
(
/
x
e
x
f
x
X
(3-32)
)
(x
fX
x
Fig. 3.9
PILLAI
18. 18
4. Gamma: if (Fig. 3.10)
If an integer
5. Beta: if (Fig. 3.11)
where the Beta function is defined as
)
,
(
G
X )
0
,
0
(
otherwise.
0,
,
0
,
)
(
)
(
/
1
x
e
x
x
f
x
X
(3-33)
n
)!.
1
(
)
(
n
n
)
,
( b
a
X )
0
,
0
(
b
a
otherwise.
0,
,
1
0
,
)
1
(
)
,
(
1
)
(
1
1
x
x
x
b
a
x
f
b
a
X (3-34)
)
,
( b
a
1
0
1
1
.
)
1
(
)
,
( du
u
u
b
a b
a
(3-35)
x
)
(x
fX
x
Fig. 3.11
1
0
)
(x
fX
Fig. 3.10
PILLAI
19. 19
6. Chi-Square: if (Fig. 3.12)
Note that is the same as Gamma
7. Rayleigh: if (Fig. 3.13)
8. Nakagami – m distribution:
),
(
2
n
X
)
(
2
n
).
2
,
2
/
(n
otherwise.
0,
,
0
,
)
(
2
2
2
/
2
x
e
x
x
f
x
X
(3-36)
(3-37)
,
)
( 2
R
X
x
)
(x
fX
Fig. 3.12
)
(x
fX
x
Fig. 3.13
PILLAI
otherwise.
0,
,
0
,
)
2
/
(
2
1
)
(
2
/
1
2
/
2
/
x
e
x
n
x
f
x
n
n
X
2
2 1 /
2
, 0
( ) ( )
0 otherwise
X
m
m mx
m
x e x
f x m
(3-38)
20. 20
9. Cauchy: if (Fig. 3.14)
10. Laplace: (Fig. 3.15)
11. Student’s t-distribution with n degrees of freedom (Fig 3.16)
.
,
1
)
2
/
(
2
/
)
1
(
)
(
2
/
)
1
(
2
t
n
t
n
n
n
t
f
n
T
,
)
,
(
C
X
.
,
)
(
/
)
( 2
2
x
x
x
fX
.
,
2
1
)
( /
|
|
x
e
x
f x
X
)
(x
fX
x
Fig. 3.14
(3-41)
(3-40)
(3-39)
x
)
(x
fX
Fig. 3.15
t
( )
T
f t
Fig. 3.16
PILLAI
21. 21
12. Fisher’s F-distribution
/ 2 / 2 / 2 1
( ) / 2
{( )/ 2}
, 0
( ) ( / 2) ( / 2) ( )
0 otherwise
m n m
m n
z
m n m n z
z
f z m n n mz
(3-42)
PILLAI
22. 22
Discrete-type random variables
1. Bernoulli: X takes the values (0,1), and
2. Binomial: if (Fig. 3.17)
3. Poisson: if (Fig. 3.18)
.
)
1
(
,
)
0
( p
X
P
q
X
P
(3-43)
),
,
( p
n
B
X
.
,
,
2
,
1
,
0
,
)
( n
k
q
p
k
n
k
X
P k
n
k
(3-44)
,
)
(
P
X
.
,
,
2
,
1
,
0
,
!
)
(
k
k
e
k
X
P
k
(3-45)
k
)
( k
X
P
Fig. 3.17
12 n
)
( k
X
P
Fig. 3.18 PILLAI
23. 23
4. Hypergeometric:
5. Geometric: if
6. Negative Binomial: ~ if
7. Discrete-Uniform:
We conclude this lecture with a general distribution due
PILLAI
(3-49)
(3-48)
(3-47)
.
,
,
2
,
1
,
1
)
( N
k
N
k
X
P
),
,
( p
r
NB
X
1
( ) , , 1, .
1
r k r
k
P X k p q k r r
r
.
1
,
,
,
2
,
1
,
0
,
)
( p
q
k
pq
k
X
P k
)
( p
g
X
, max(0, ) min( , )
( )
m N m
k n k
N
n
m n N k m n
P X k
(3-46)
24. 24
PILLAI
to Polya that includes both binomial and hypergeometric as
special cases.
Polya’s distribution: A box contains a white balls and b black
balls. A ball is drawn at random, and it is replaced along with
c balls of the same color. If X represents the number of white
balls drawn in n such draws, find the
probability mass function of X.
Solution: Consider the specific sequence of draws where k
white balls are first drawn, followed by n – k black balls. The
probability of drawing k successive white balls is given by
Similarly the probability of drawing k white balls
0, 1, 2, , ,
X n
2 ( 1)
2 ( 1)
W
a a c a c a k c
p
a b a b c a b c a b k c
(3-50)
25. 25
PILLAI
followed by n – k black balls is given by
Interestingly, pk
in (3-51) also represents the probability of
drawing k white balls and (n – k) black balls in any other
specific order (i.e., The same set of numerator and
denominator terms in (3-51) contribute to all other sequences
as well.) But there are such distinct mutually exclusive
sequences and summing over all of them, we obtain the Polya
distribution (probability of getting k white balls in n draws)
to be
n
k
1 1
0 0
( )
( 1)
( 1) ( 1)
.
k w
k n k
i j
b jc
a ic
a b ic a b j k c
b b c b n k c
p p
a b kc a b k c a b n c
(3-51)
1 1
0 0
( )
( ) , 0,1,2, , .
k n k
k
i j
n n
k k
b jc
a ic
a b ic a b j k c
P X k p k n
(3-52)
26. 26
PILLAI
Both binomial distribution as well as the hypergeometric
distribution are special cases of (3-52).
For example if draws are done with replacement, then c = 0
and (3-52) simplifies to the binomial distribution
where
Similarly if the draws are conducted without replacement,
Then c = – 1 in (3-52), and it gives
( ) , 0,1,2, ,
k n k
n
k
P X k p q k n
(3-53)
, 1 .
a b
p q p
a b a b
( 1)( 2) ( 1) ( 1) ( 1)
( )
( )( 1) ( 1) ( ) ( 1)
n
k
a a a a k b b b n k
P X k
a b a b a b k a b k a b n
27. 27
which represents the hypergeometric distribution. Finally
c = +1 gives (replacements are doubled)
we shall refer to (3-55) as Polya’s +1 distribution. the general
Polya distribution in (3-52) has been used to study the spread
of contagious diseases (epidemic modeling).
1 1
1
( 1)! ( 1)! ( 1)! ( 1)!
( )
( 1)! ( 1)! ( 1)! ( 1)!
= .
n
k
a k b n k
k n k
a b n
n
a k a b b n k a b k
P X k
a a b k b a b n
(3-55)
! !( )! !( )!
( )
!( )! ( )!( )! ( )!( )!
a b
k n k
a b
n
n a a b k b a b n
P X k
k n k a k a b b n k a b k
(3-54)
PILLAI