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
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
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
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
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
choo
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 proba
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
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
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. Samp
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
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 stag
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
det
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
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 sam
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
a
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
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 ther
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
watch
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
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
watch
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
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/sev
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
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 confidenc
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
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 probabi
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 interest
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 possib
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 esti