Question
Show that
u=Ce
x + ( 2 ) y
is a solution to
u x +u y 2u=0
Solution
u x +u y 2u=0
u=Ce
x + ( 2 ) y
u x =C e x +(2 ) y
u y =C ( 2 ) e
x+ ( 2 ) y
=2 Ce x+(2 ) y Cex +(2 ) y
Substituting into the PDE, we obtain:
Ce x+(2 ) y +2 Ce x+(2 ) y Ce x+(2 ) y
Solve the following using separation of variables:
u x +u y 2u=0
u= XY
Let
be the solution
u x =X ' Y
Then,
u y = XY '
The above must satisfy the given PDE.
'
'
X Y + X Y 2 XY =0
Dividing through by
X' Y '
+ 2=0
X Y
'
'
X
Y
=2 =
X
Y
X'
=
X
Either
Or
2
Y'
Solution by Kudzordzi
Double integrals
1 1
x 5 y 3 e xy dydx
0 0
Integrating wrt y first
Let u=xy du=xdy
u
Also y= x
By substitution and change of limit
1 1
1 1
x 5 y 3 e xy dydx= x 4 y 3 e xy x dy dx
0 0
0 0
1 x
x 4
0 0
u3 u
e dudx
3
x
1 x
x u3 e u
Sampling Design and Analysis
MTH 494
Two-stage Cluster Sampling
2
3
4
5
6
Review of The Course
7
8
Basics of Statistics
Definition: Science of collection, presentation, analysis, and reasonable
interpretation of data.
Statistics presents a rigorous scient
Sampling Design and Analysis
MTH 494
Neymans allocation
In some stratified sampling problems the
cost of obtaining an observation is the
same for all strata.
If costs are unknown, we may be willing to
assume that the costs per observation are
equal.
If c1
Sampling Design and Analysis
MTH 494
Comparisons among the
designs and methods
With an array of sampling designs and
methods of analysis available, we now
summarize earlier discussions on how one
chooses an appropriate design for a
particular problem.
Sim
Sampling Design and Analysis
MTH 494
Sampling with unequal
probabilities
Up to now, we have only discussed
sampling
schemes
in
which
the
probabilities of choosing sampling units
are equal.
Equal probabilities give schemes that are
often easy to design and
Sampling Design and Analysis
MTH 494
CLUSTER SAMPLING
2
Definition: A cluster sample is a simple
random sample in which each sampling
unit is a collection, or cluster, of elements.
Cluster sampling is less costly than simple
or stratified random sampling
Sampling Design and Analysis
MTH 494
Selecting sample size for
Estimating Population Means
The quantity of information in a cluster
and Totals
sample is affected by two factors, the
number of clusters and the relative cluster
size.
We have not encountered
Sampling Design and Analysis
MTH 494
Systematic Sampling
As we have seen in previous units , both
simple and stratified random sampling
require very detailed work in the sample
selection process. Sampling units on an
adequate frame must be numbered (or
ot
Sampling Design and Analysis
MTH 494
Cluster sampling with
probabilities proportional to size
We observed in previous discussions that
we can sometimes reduce the variance of
an estimator by sampling units with
probabilities proportional to a measure of
t
Sampling Design and Analysis
MTH 494
Selecting sample sizes
The problem of choosing the sample sizes
in two-stage sampling is much more
difficult that in previous chapter, which only
involved one stage of sampling.
We have to select values for n and all t
Sampling Design and Analysis
MTH 494
Estimation of population
proportion
An investigator frequently wishes to use
data from a systematic sample to estimate
a population proportion. For example, to
determine the proportion of registered
voters in favor of
Sampling Design and Analysis
MTH 494
Estimators
Ratio estimator of a population mean y:
y :
Estimated variance of
2
Bound on the error of estimation:
Note that we dont need to know x or N to
estimate y when using the ratio
procedure; however we must know
Sampling Design and Analysis
MTH 494
Systematic
Sampling
2
Session Objectives
To introduce basic sampling concepts
in systematic sampling
Demonstrate how to select a random
sample using systematic sampling
design
Estimation of different parameters in
syst
Sampling Design and Analysis
MTH 494
The ratio technique for estimating a
population total y was applied estimating
the total sugar content of a truckload of
y
oranges. The simple estimator N
is not
applicable because we dont know N, the
total number of o
Sampling Design and Analysis
MTH 494
Survey that require the use of
Ratio Estimation
Estimation a population total sometimes
requires the use of subsidiary variables.
Let us take an example to explain this
situation.
2
Understanding example
The wholesale
Sampling Design and Analysis
MTH 494
Review
2
Regression Estimation
We observed that the ratio estimator is
most appropriate when the relationship
between y and x is linear through the
origin.
If there is evidence of a linear relationship
between the obse
Sampling Design and Analysis
MTH 494
Estimation of a Population
Proportion
In our numerical example, we have been
interested in estimating the average or the
total number of hours per week spent
watching television.
In contrast, suppose that the advertisi
Sampling Design and Analysis
MTH 494
How to draw a Stratified
Random Sample
First step is to clearly specify the strata.
Place each sampling unit of population in
its appropriate stratum.
Not an easy
task
After the sampling units are divided into
strata,
Sampling Design and Analysis
MTH 494
Estimation of a Population
Proportion
In our numerical example, we have been
interested in estimating the average or the
total number of hours per week spent
watching television.
In contrast, suppose that the advertisi
Sampling Design and Analysis
MTH 494
How to draw a Stratified
Random Sample
First step is to clearly specify the strata.
Place each sampling unit of population in
its appropriate stratum.
Not an easy
task
After the sampling units are divided into
strata,
Sampling Design and Analysis
MTH 494
STRATIFIED SAMPLING
2
STRATIFIED SAMPLING
1. Stratification: The elements in the
population are divided into layers/groups/
strata based on their values on one/several
auxiliary variables. The strata must be nonoverlap
Sampling Design and Analysis
MTH 494
LECTURE-6
2
So far we have done.
3
Sampling
Techniques
Important statistical terms
Population:
a set which includes all
measurements of interest
to the researcher
(The collection of all
measurements, or counts
that are
Sampling Design and Analysis
MTH 494
Confidence Intervals
Confidence Interval: An interval of values computed
from the sample, that is almost sure to cover the true
population value.
We make confidence intervals using values computed from the
sample, not
Sampling Design and Analysis
MTH 494
Review
2
Confidence Intervals
Confidence Interval: An interval of values computed
from the sample, that is almost sure to cover the true
population value.
We make confidence intervals using values computed from the
sam
Sampling Design and Analysis
MTH 494
Review
2
Sampling with probabilities
proportional to size
So far we have discussed all cases
depended on samples being a simple
random sample.
In real life probabilities cannot be same for
all samples.
Varying the prob
Sampling Design and Analysis
MTH 494
Review
2
Estimation of a Population
Proportion
First have a look what does proportion
mean?
3
Estimation of a Population
Proportion
Researchers frequently interested in the
portion of population possessing a
specified
Sampling Design and Analysis
MTH 494
LECTURE-7
2
Simple Random Sampling
(SRS)
Simplest sampling design
Def-1: If a sample of size n is drawn from
a population of size N in such a way that
every possible sample of size n has the
same chance (probability) o
Sampling Design and Analysis
MTH 494
Review of previous stuff
2
How to draw a SR Sample
This is not as difficult as it looks
But selection is important because it leads
to
Investigator bias
Poor estimation
The procedure for selecting a Simple
Random Samp