The sampling distribution of the mean describes the probability distribution of sample means that would be obtained by drawing all possible random samples of a given size from a population. For a random sample of size n drawn from a normal population with mean μ and standard deviation σ, the sampling distribution of the mean is a normal distribution with mean μ and standard deviation σ/√n. When the population standard deviation σ is unknown, the t distribution provides the sampling distribution of the mean if the population is normal. The t distribution approaches the normal distribution as the degrees of freedom increase.
Chapter 5 part1- The Sampling Distribution of a Sample Meannszakir
Mathematics, Statistics, Population Distribution vs. Sampling Distribution, The Mean and Standard Deviation of the Sample Mean, Sampling Distribution of a Sample Mean, Central Limit Theorem
1. continuous probability distribution
2. Normal Distribution
3. Application of Normal Dist
4. Characteristics of normal distribution
5.Standard Normal Distribution
Chapter 5 part1- The Sampling Distribution of a Sample Meannszakir
Mathematics, Statistics, Population Distribution vs. Sampling Distribution, The Mean and Standard Deviation of the Sample Mean, Sampling Distribution of a Sample Mean, Central Limit Theorem
1. continuous probability distribution
2. Normal Distribution
3. Application of Normal Dist
4. Characteristics of normal distribution
5.Standard Normal Distribution
Basics of probability in statistical simulation and stochastic programmingSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 2.
More info at http://summerschool.ssa.org.ua
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
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This 10 hours class is intended to give students the basis to empirically solve statistical problems. Talk 1 serves as an introduction to the statistical software R, and presents how to calculate basic measures such as mean, variance, correlation and gini index. Talk 2 shows how the central limit theorem and the law of the large numbers work empirically. Talk 3 presents the point estimate, the confidence interval and the hypothesis test for the most important parameters. Talk 4 introduces to the linear regression model and Talk 5 to the bootstrap world. Talk 5 also presents an easy example of a markov chains.
All the talks are supported by script codes, in R language.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 3.
More info at http://summerschool.ssa.org.ua
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
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1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
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Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
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The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
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Bob Boule
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Gopinath Rebala
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Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
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The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
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1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
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Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
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UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
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Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
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Charlie Greenberg, Host
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JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
2. Contents:
Sampling Distributions
1. Populations and Samples
2. The Sampling Distribution of the mean ( known)
3. The Sampling Distribution of the mean ( unknown)
4. The Sampling Distribution of the Variance
4. Populations and Samples
Population: A set or collection of all the objects, actual
or conceptual and mainly the set of numbers,
measurements or observations which are under
investigation.
Finite Population : All students in a College
Infinite Population : Total water in the sea or all the
sand particle in sea shore.
Populations are often described by the distributions of
their values, and it is common practice to refer to a
population in terms of its distribution.
5. Finite Populations
• Finite populations are described by the actual
distribution of its values and infinite populations are
described by corresponding probability distribution or
probability density.
• “Population f(x)” means a population is described
by a frequency distribution, a probability distribution
or a density f(x).
6. Infinite Population
If a population is infinite it is impossible to observe all
its values, and even if it is finite it may be impractical
or uneconomical to observe it in its entirety. Thus it is
necessary to use a sample.
Sample: A part of population collected for investigation
which needed to be representative of population and
to be large enough to contain all information about
population.
7. Random Sample (finite population):
• A set of observations X1, X2, …,Xn constitutes a
random sample of size n from a finite
population of size N, if its values are chosen so
that each subset of n of the N elements of the
population has the same probability of being
selected.
8. Random Sample (infinite Population):
A set of observations X1, X2, …, Xn constitutes a random sample
of size n from the infinite population ƒ(x) if:
1. Each Xi is a random variable whose distribution is given
by ƒ(x)
2. These n random variables are independent.
We consider two types of random sample: those drawn with
replacement and those drawn without replacement.
9. Sampling with replacement:
• In sampling with replacement, each object
chosen is returned to the population before the
next object is drawn. We define a random
sample of size n drawn with replacement, as
an ordered n-tuple of objects from the
population, repetitions allowed.
10. The Space of random samples drawn with
replacement:
• If samples of size n are drawn with replacement from a
population of size N, then there are Nn such samples. In any
survey involving sample of size n, each of these should have
same probability of being chosen. This is equivalent to making
a collection of all Nn samples a probability space in which each
sample has probability of being chosen 1/Nn.
• Hence in above example There are 32 = 9 random samples of
size 2 and each of the 9 random sample has probability 1/9 of
being chosen.
11. Sampling without replacement:
In sampling without replacement, an object chosen is not returned
to the population before the next object is drawn. We define
random sample of size n, drawn without replacement as an
unordered subset of n objects from the population.
12. The Space of random samples drawn
without replacement:
• If sample of size n are drawn without
replacement from a population of size N, then
N
there are
n
such samples. The collection of
all random samples drawn without
replacement can be made into a probability
space in which each sample has same chance
of being selected.
13. Mean and Variance
If X1, X2, …, Xn constitute a random sample, then
n
X i
i 1
X
n
is called the sample mean and
n
(X i X )
2
i 1
2
S
n 1
is called the sample variance.
14. Sampling distribution:
• The probability distribution of a random variable
defined on a space of random samples is called a
sampling distribution.
15. The Sampling Distribution of the Mean ( Known)
Suppose that a random sample of n observations has been taken from
some population and x has been computed, say, to estimate the mean
of the population. If we take a second sample of size n from this
population we get some different value for x . Similarly if we take
several more samples and calculate x , probably no two of the x ' s .
would be alike. The difference among such x ' s are generally attributed
to chance and this raises important question concerning their
distribution, specially concerning the extent of their chance of
fluctuations.
Let X and
2
X
be mean and variance for sampling distributi on of the
mean X .
16. The Sampling Distribution of the Mean ( Known)
2
Formula for μ X and σ X :
Theorem 1: If a random sample of size n is taken from a population
having the mean and the variance 2, then X is a random variable
whose distribution has the mean .
2
For samples from infinite populations the variance of this distribution is .
n
N n
2
For samples from a finite population of size N the variance is n .
N 1
2
(for infinite Population )
n
That is X and
2
X
N n
2
(for finite Population )
n N 1
17. The Sampling Distribution of the Mean ( Known)
Proof of for infinite population for the
X
continuous
case: From the definition we have
X ..... x f ( x1 , x 2 ,...., x n )dx 1 dx 2 .... dx n
n
xi
..... n
f ( x1 , x 2 ,...., x n )dx 1 dx 2 .... dx n
i 1
n
1
n
..... x i
f ( x1 , x 2 ,...., x n )dx 1 dx 2 .... dx n
i 1
18. The Sampling Distribution of the Mean ( Known
Where f(x1, x2, … , xn) is the joint density function of the random variables
which constitute the random sample. From the assumption of random sample
(for infinite population) each Xi is a random variable whose density function
is given by f(x) and these n random variables are independent, we can write
f(x1, x2, … , xn) = f(x1) f(x2) …… f(xn)
and we have
n
1
X
n
..... x i
f ( x1 ) f ( x 2 ).... f ( x n )dx 1 dx 2 .... dx n
i 1
n
1
n
f ( x1 ) dx 1 ... x i f ( x i ) dx i .... f ( x n ) dx n
i 1
19. The Sampling Distribution of the Mean ( Known)
Since each integral except the one with the integrand xi f(xi) equals 1 and the
one with the integrand xi f(xi) equals to , so we will have
n
1 1
X
n
n
n .
i 1
Note: for the discrete case the proof follows the same steps, with integral sign
replaced by ' s.
For the proof of X 2 / n we require the following result
2
Result: If X is a continuous random variable and Y = X - X , then Y = 0
and hence Y2 X .
2
Proof: Y = E(Y) = E(X - X) = E(X) - X = 0 and
Y2 = E[(Y - Y)2] = E [((X - X ) - 0)2] = E[(X - X)2] = X2.
20. The Sampling Distribution of the Mean ( Known)
• Regardless of the form of the population distribution, the
distribution of is approximately normal with mean and
variance 2/n whenever n is large.
• In practice, the normal distribution provides an excellent
X
approximation to the sampling distribution of the mean
for n as small as 25 or 30, with hardly any restrictions on the
shape of the population.
• If the random samples come from a normal population, the
X
sampling distribution of the mean is normal regardless of
the size of the sample.
21. The Sampling Distribution of the mean
( unknown)
• Application of the theory of previous section requires knowledge of the
population standard deviation .
• If n is large, this does not pose any problems even when is unknown, as
it is reasonable in that case to use for it the sample standard deviation s.
• However, when it comes to random variable whose values are given by
very little is known about its exact sampling distribution for
small values of n unless we make the assumption that the sample comes
from a normal population.
X
,
S / n
22. The Sampling Distribution of the mean
( unknown)
Theorem : If is the mean of a random sample of size n taken from a
normal population having the mean and the variance 2, and
X
(Xi X )
n 2
, then
2
S
i 1 n 1
X
t
S/ n
is a random variable having the t distribution with the parameter = n – 1.
This theorem is more general than Theorem 6.2 in the sense that it does not
require knowledge of ; on the other hand, it is less general than Theorem 6.2
in the sense that it requires the assumption of a normal population.
23. The Sampling Distribution of the mean
( unknown)
• The t distribution was introduced by William S.Gosset in
1908, who published his scientific paper under the pen name
“Student,” since his company did not permit publication by
employees. That’s why t distribution is also known as the
Student-t distribution, or Student’s t distribution.
• The shape of t distribution is similar to that of a normal
distribution i.e. both are bell-shaped and symmetric about the
mean.
• Like the standard normal distribution, the t distribution has the
mean 0, but its variance depends on the parameter , called the
number of degrees of freedom.
24. The Sampling Distribution of the mean
( unknown)
t ( =10)
Normal
t ( =1)
Figure: t distribution and standard normal distributions
25. The Sampling Distribution of the mean
( unknown)
• When , the t distribution approaches the standard
normal distribution i.e. when , t z.
• The standard normal distribution provides a good
approximation to the t distribution for samples of size 30 or
more.