Quantitative Techniques
1.
Discuss the various mathematical models.
A
mathematical model
is a description of a
system
using
mathematical
concepts and language. The
process of developing a mathematical model is termed
mathematical modeling
. Mathematical models
are used not only in the
natural sciences
(such as
physics
,
biology
,
earth science
,
meteorology
)
and
engineering
disciplines (e.g.
computer science
,
artificial intelligence
)
, but also in the
social
sciences
(such as
economics
,
psychology
,
sociology
and
poli
tical
science
);
physicists
,
engineers
,
sta
tisticians
,
operations research
analysts and
economists
use
mathematical models most extensively
. A model may help to explain a system and to study the effects of
different components, and to make predictions about behaviour.
Mathematical models can take many forms, including but not limited to
dynamical systems
,
statistical
models
,
diffe
rential equations
, or
game theoretic models
. These and other types of models can overlap,
with a given model involving a variety of abstract structures. In general, mathematical
models may
include
logical models
, as far as logic is taken as a part of mathematics. In many cases, the quality of a
scientific field depends on how well the mathematical models
developed on the theoretical side agree
with results of repeatable experiments. Lack of agreement between theoretical mathematical models and
experimental measurements often leads to important advances as better theories are developed.
Many everyday activ
ities carried out without a thought are uses of mathematical models. A
geographical
map projection
of a region of the earth onto a small, plane surface is a model
[1]
which
can be used for many purposes such as planning travel.
Another simple activity is predicting the position of a vehicle from its initial position, direction and
speed of travel,
using the equation that distance travelled is the product of time and speed. This is
known as
dead reckoning
when used more formally. Mathematical modelling in this way does no
t
necessarily require formal mathematics; animals have been shown to use dead reckoning
[2]
[3]
.
Population
Growth
. A simple (though approximate) model of population growth is the
Malthusian
growth model
. A slightly more realistic and largely used population growth model is the
logistic
function
, and its extensions.
Model of a particle in a potenti
al

field
. In this model we consider a particle as being a point of mass
which describes a trajectory in space which is modeled by a function giving its coordinates in space
as a function of time. The potential field is given by a function
V
:
R
3
→
R
and th
e trajectory is a
solution of the
differential equation
Note this model assumes the particle is a point mass, which is certainly known to be false in
many cases
in which we use this model; for example, as a model of planetary motion.
Model of rational behavior for a consumer
. In this model we assume a consumer faces a
choice of
n
commodities labeled 1,2,...,
n
each with a market price
p
1
,
p
2
,...,
p
n
. The
consumer
is assumed to have a
cardinal
utility function
U
(cardinal in the sense that it
assigns numerical values to utilities), depending on the amounts of
commodities
x
1
,
x
2
,...,
x
n
consumed. The model further assumes that the consumer has a
budget
M
which is
used to purchase a vector
x
1
,
x
2
,...,
x
n
in such a way as to
maximize
U
(
x
1
,
x
2
,...,
x
n
). The problem of rational behavior in this model then becomes
an
o
ptimization
problem, that is:
subject to:
2.
Explain
the various methods to find initial solution for
transpiration
problem with an
example.
transpiration problems (sweaty armpits, sweaty
hands)
The sweat glands of excessively sweaty underarms cannot be neutralised by deodorants
indefinitely. After all, some ingredients in deodorants blocking the pores, are not well tolerated
by everyone.
temporary treatment
Sweat secretion can be diminished or s
topped for a few months by local injection of
Botulinum
toxin
(brand names: Dysport®, Botox®). Especially for people with very excessive armpit
perspiration with or without odour
problems, this treatment can be very effective. The average
duration of effectiveness is six to eight months.
These treatments can be given during consultation. Injections for sweaty armpits are usually well
tolerated, if not, local anaesthetic can fairly
simply be administered without directly influencing
dexterity.
As the palms of the hands and soles of the feet are very sensitive, a local anaesthetic is required
to treat sweaty hands and feet. As a result, it is impossible to drive a car for several ho
urs after
treatment.
permanent, surgical diminishing of armpit transpiration
Through an incision of one centimetre at most, under local anaesthesia, the skin of the armpits
can be released from the underlying fat. The sweat glands are scraped from the und
ersurface of
the skin by modified liposculpture. Admission is limited to a few hours. There is some discomfort,
but light work can be resumed the next day. The skin wil adhere to the fat layer like a graft.
Usually, healing is uneventful. Because of the ve
ry short incision there is much less visible scarring
than after open surgery on the armpits. Temporary hardening of the operated zone is normal.
There is a small chance that a complication will arise by which the skin does not completely take,
resulting i
n an open wound. This does require appropriate dressings and wound care for
considerable time.
3.
What is normal distribution? Explain the properties of normal distribution.
In
probability theory
, the
normal
(or
Gaussian
)
distribution
is a
continuous probability distribution
that
has a bell

shaped
probability density function
, known as the
Gaussian function
or informally the
bell
curve
:
[nb 1]
where parameter
μ
is the
mean
or
expectation
(location of the peak) and
σ
2
is the variance.
σ
is
known as the
standard deviation
. The distribution with
μ
= 0
and
σ
2
= 1
is called the
standard
normal
. A normal distribution is often used as a
first approximation to describe real

valued
random
variables
that cluster around a single
mean
value.
The normal
distribution is considered the most prominent probability distribution in
statistics
. There
are several reasons for this:
[1]
First, the normal distribution is very tractable analytically, that is, a
large number of results involving this distribution can be derived in explicit form. Second, the normal
distribution arises as the outcome of the
central limit theorem
, which states that under mild conditions
the sum of a large number of
random variables
is distributed approximately normally. Finally, the
"bell" shape of the normal distribution makes it a convenient choice for modelling a large variety of
random variables encountered in practice.
For this reason, the normal distribution i
s commonly encountered in practice, and is used throughout
statistics,
natural sciences
, and
social sciences
[2]
as a simple model for complex phenomena. For
example, the
observa
tional error
in an experiment is usually assumed to follow a normal distribution,
and the
propagation of uncertainty
is computed using this assumption.
Note that a normally

distributed variable has a symmetric distribution about its mean. Quantities that
grow exponentially
,
such as prices, incomes or populations, are o
ften
skewed to the right
, and hence may be better
described by other distributions, such as the
log

normal distribution
or
Pareto distribution
. In addition,
the probability of seeing a normally

distributed value that is far (i.e. more than a few
standard
deviations
) from the mean drops off extremely rapidly. As a result,
statist
ical inference
using a normal
distribution is not robust to the presence of
outliers
(data that is unexpectedly far from the mean, due
to exceptional circumstances, observational error,
etc.). When outliers are expected, data may be
better described using a
heavy

tailed
distribution such as the
Student's t

distribution
.
From a technical perspective, alternative characterizations are possible, for example:
The normal distribution is the only
absolutely continuous
distribution all of
whose
cumulants
beyond the first two (i.e. other than the
mean
and
variance
) are zero.
For a given mean and variance, the corresponding normal distribution is the continuous
distribution with the
maximum entropy
.
[3]
[4]
The normal distributions are a sub

class of the
elliptical distributions
.
4.
What are the assumptions made in a waiting line method?
The
method of lines
(MO
L, NMOL, NUMOL) (
Schiesser, 1991
;
Hamdi, et al., 2007
;
Schiesser, 2009
)
is a technique for solving
partial differential equat
ions
(PDEs) in which all but one dimension is
discretized. MOL allows standard, general

purpose methods and software, developed for the numerical
integration of ODEs and DAEs, to be used. A large number of integration routines have been developed
over the
years in many different programming languages, and some have been published as
open
source
resources; see for example
Lee and Schiesser (2004)
.
The method of lines most often refers to the construction or analysis of numerical methods for partial
differential equations that proceeds by first discretizing the spatial derivatives only and leaving the time
variable
continuous. This leads to a system of ordinary differential equations to which a numerical method
for initial value ordinary equations can be applied. The method of lines in this context dates back to at
least the early 1960s
Sarmin and Chudov
. Many papers discussing the accuracy and stability of the
method of lines for various types of partial differential equations have appeared since (for
example
Zafarullah
or
Verwer and Sanz

Serna
). W. E. Schiesser of
Lehigh University
is one of the major
proponents of the method of lines, having published widely in this field.
[
edit
]
Application to elliptical equations
MOL requires that the PDE problem is well

posed as an initial valu
e (
Cauchy
) problem in at least one
dimension, because ODE and DAE integrators are
initial value problem
(IVP) solvers.
Thus it cannot be used directly on purely elliptic equations, such as
Laplace's equation
. However, MOL
has been used to solve Lap
lace's equation by using the
method of false transients
(
Schiesser,
1991
;
Schiesser, 1994
). In this method, a
time derivative of the dependent variable is added to Laplace’s
equation. Finite differences are then used to approximate the spatial derivatives, and the resulting system
of equations is solved by MOL. It is also possible to solve elliptical problems by a
semi

analytical method
of lines
(
Subramanian, 2004
). In this method the discretization process results in a set of ODE's that are
solved by exploiting properties of the associated ex
ponential matrix. For a sample code,
visit
http://www.maple.eece.wustl.edu
.
5.
What does ‘significant’ mean
In
statistics
, a result is
called "statistically significant" if it is unlikely to have occurred by
chance
. The
phrase
test of significance
was coined by
Ronald Fisher
.
[1]
As used
in statistics,
significant
does not
mean
important
or
meaningful
, as it does in everyday speech. Research analysts who focus solely on
significant results may miss important response patterns which individually may fall under the threshold
set for tests o
f significance. Many researchers urge that tests of significance should always be
accompanied by
effect

size
statistics, which approximate the size and thus the practical importance
of the
difference.
The amount of evidence required to accept that an event is unlikely to have arisen by chance is known as
the
significance level
or critical
p

value
: in traditional
Fisherian
statistical hypothesis testing
, the p

value
is the proba
bility of observing data at least as extreme as that observed,
given that the null hypothesis is
true
. If the obtained p

value is small then it can be said either the
null hy
pothesis
is false or an unusual
event has occurred. P

values do not have any repeat sampling interpretation.
[
citation needed
][
clarification needed
]
An alternative (but nevertheless related) statistical hypothesis testing framework is the
Neyman
–
Pearson
frequentist school which requires both a null and an alternative hypothesis to be defined and
investigates the repeat sampling properties of the procedure, i.e. the probability that a dec
ision to reject
the null hypothesis will be made when it is in fact true and should not have been rejected (this is called a
"false positive" or
Type I error
) and the probability t
hat a decision will be made to accept the null
hypothesis when it is in fact false (
Type II error
). Fisherian p

values are philosophically different from
Neyman
–
Pearson Type I er
rors. This confusion is unfortunately propagated by many statistics
textbooks.
[2]
6.
Explain the structure of M/M/1 Model for infinite population.
7.
Why is sampling
important?
Application to probabilistic inference
Such methods are frequently used to estimate posterior densities or expectations in state and/or
parameter estimation problems in probabilistic models that are too hard to treat analytically, for example
in
Bayesian networks
.
[
edi
t
]
Application to simulation
Importance sampling
is a
variance reduction
technique that can be used in the
Monte Carlo method
.
The idea behind importance sampling is that certain values of the input
random variables
in
a
simulation
have more impact on the parameter being estimated than others. If these "important" values
are emphasized by sampling more frequently, then the
estimator
variance can be reduced. Hence, the
basic methodology in importance sampling is to choose a distribution which "encourages" the important
values. This use of "biased" distributio
ns will result in a biased estimator if it is applied directly in the
simulation. However, the simulation outputs are weighted to correct for the use of the biased distribution,
and this ensures that the new importance sampling estimator is unbiased. The w
eight is given by
the
likelihood ratio
, that is, the
Radon
–
Nikodym derivative
of the true underlying distribution with respect
to the biased simulation distribution.
The fundamental issue in implementing importance sampling simulation is the choice of the biased
distribution which encourages the important regions
of the input variables. Choosing or designing a good
biased distribution is the "art" of importance sampling. The rewards for a good distribution can be huge
run

time savings; the penalty for a bad distribution can be longer run times than for a general Mo
nte Carlo
simulation without importance sampling.
[
edit
]
Mathematical approach
Consider estimating by simulation the
probability
of an event
, where
X
is a random variable
with
distribution
F
and
probability density function
, where prime denotes
derivative
.
A
K

length
independent and identically distributed
(i.i.d.) sequence
is generated from the
distribution
F
, and the number
k
t
of random vari
ables that lie above the threshold
t
are counted. The
random variable
k
t
is characterized by the
Binomial distribution
One can show that
, and
, so in the
limit
we are able to obtain
p
t
. Note that the variance is low if
. Importance
sampling is concerned with the determination and use of an alternate density function
(for X),
usually referred to as a biasing density, for the simulation experiment. This d
ensity allows the
event
to occur more frequently, so the sequence lengths
K
gets smaller for a
given
estimator
variance. Alternatively, for a given
K
, use of the biasing density result
s in a variance
smaller than that of the conventional Monte Carlo estimate. From the definition of
, we can
introduce
as below.
where
is
a likelihood ratio and is referred to as the weighting function. The last equality in the above
equation motivates the estimator
This is the importance sampling estimator of
and is unbiased. That is, the estimation
procedure is to generate i.i.d. samp
les from
and for each sample which exceeds
,
the estimate is incremented by the weight
evaluated at the sample value. The
results are averaged over
trials. The variance of the importance sampling estimator
is easily shown to be
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