This chapter discusses several key continuous probability distributions including the uniform, normal, and exponential distributions. It covers the characteristics and formulas for computing probabilities for each distribution. For the uniform distribution, it discusses how to calculate the mean, find probabilities, and graph the distribution. For the normal distribution, it outlines the characteristics of the bell curve shape and covers converting to the standard normal distribution and using the empirical rule. It also provides examples of finding probabilities for observations within given ranges of a normal distribution.
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Chapter 6: Normal Probability Distribution
6.1: The Standard Normal Distribution
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Chapter 2: Exploring Data with Tables and Graphs
2.4: Scatterplots, Correlation, and Regression
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Chapter 6: Normal Probability Distribution
6.1: The Standard Normal Distribution
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Chapter 2: Exploring Data with Tables and Graphs
2.4: Scatterplots, Correlation, and Regression
Chapter 5 part2- Sampling Distributions for Counts and Proportions (Binomial ...nszakir
Mathematics, Statistics, Sampling Distributions for Counts and Proportions, Binomial Distributions for Sample Counts,
Binomial Distributions in Statistical Sampling, Binomial Mean and Standard Deviation, Sample Proportions, Normal Approximation for Counts and Proportions, Binomial Formula
Turning from discrete to continuous distributions, in this section we discuss the normal distribution. This is the most important continuous distribution because in applications many random variables are normal random variables (that is, they have a normal distribution) or they are approximately normal or can be transformed into normal random variables in a relatively simple fashion. Furthermore, the normal distribution is a useful approximation of more complicated distributions, and it also occurs in the proofs of various statistical tests.
Normal Distribution, also called Gaussian Distribution, is one of the widely used continuous distributions existing which is used to model a number of scenarios such as marks of students, heights of people, salaries of working people etc.
Each binomial distribution is defined by n, the number of trials and p, the probability of success in any one trial.
Each Poisson distribution is defined by its mean.
In the same way, each Normal distribution is identified by two defining characteristics or parameters: its mean and standard deviation.
The Normal distribution has three distinguishing features:
• It is unimodal, in other words there is a single peak.
• It is symmetrical, one side is the mirror image of the other.
• It is asymptotic, that is, it tails off very gradually on each side but the line representing the distribution never quite meets the horizontal axis
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Elementary Statistics Practice Test 2 Solutions
Chapter 4: Probability
This ppt is a part of Business Analytics course.
Normal distribution : -
The Normal Distribution, also called the Gaussian Distribution, is the most significant continuous probability distribution.
A normal distribution is a
symmetric, bell-shaped curve
that describes the distribution of continuous random variables.
The normal curve describes how data are distributed in a population.
A large number of random variables are either nearly or exactly represented by the normal distribution
The normal distribution can be used to represent a wide range of data, such as test scores, height measurements, and weights of people in a population.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.2: Real Applications of Normal Distributions
Chapter 5 part2- Sampling Distributions for Counts and Proportions (Binomial ...nszakir
Mathematics, Statistics, Sampling Distributions for Counts and Proportions, Binomial Distributions for Sample Counts,
Binomial Distributions in Statistical Sampling, Binomial Mean and Standard Deviation, Sample Proportions, Normal Approximation for Counts and Proportions, Binomial Formula
Turning from discrete to continuous distributions, in this section we discuss the normal distribution. This is the most important continuous distribution because in applications many random variables are normal random variables (that is, they have a normal distribution) or they are approximately normal or can be transformed into normal random variables in a relatively simple fashion. Furthermore, the normal distribution is a useful approximation of more complicated distributions, and it also occurs in the proofs of various statistical tests.
Normal Distribution, also called Gaussian Distribution, is one of the widely used continuous distributions existing which is used to model a number of scenarios such as marks of students, heights of people, salaries of working people etc.
Each binomial distribution is defined by n, the number of trials and p, the probability of success in any one trial.
Each Poisson distribution is defined by its mean.
In the same way, each Normal distribution is identified by two defining characteristics or parameters: its mean and standard deviation.
The Normal distribution has three distinguishing features:
• It is unimodal, in other words there is a single peak.
• It is symmetrical, one side is the mirror image of the other.
• It is asymptotic, that is, it tails off very gradually on each side but the line representing the distribution never quite meets the horizontal axis
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 2 Solutions
Chapter 4: Probability
This ppt is a part of Business Analytics course.
Normal distribution : -
The Normal Distribution, also called the Gaussian Distribution, is the most significant continuous probability distribution.
A normal distribution is a
symmetric, bell-shaped curve
that describes the distribution of continuous random variables.
The normal curve describes how data are distributed in a population.
A large number of random variables are either nearly or exactly represented by the normal distribution
The normal distribution can be used to represent a wide range of data, such as test scores, height measurements, and weights of people in a population.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.2: Real Applications of Normal Distributions
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
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...
Speakers:
Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
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).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
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Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
Search and Society: Reimagining Information Access for Radical FuturesBhaskar Mitra
The field of Information retrieval (IR) is currently undergoing a transformative shift, at least partly due to the emerging applications of generative AI to information access. In this talk, we will deliberate on the sociotechnical implications of generative AI for information access. We will argue that there is both a critical necessity and an exciting opportunity for the IR community to re-center our research agendas on societal needs while dismantling the artificial separation between the work on fairness, accountability, transparency, and ethics in IR and the rest of IR research. Instead of adopting a reactionary strategy of trying to mitigate potential social harms from emerging technologies, the community should aim to proactively set the research agenda for the kinds of systems we should build inspired by diverse explicitly stated sociotechnical imaginaries. The sociotechnical imaginaries that underpin the design and development of information access technologies needs to be explicitly articulated, and we need to develop theories of change in context of these diverse perspectives. Our guiding future imaginaries must be informed by other academic fields, such as democratic theory and critical theory, and should be co-developed with social science scholars, legal scholars, civil rights and social justice activists, and artists, among others.
2. Learning Objectives
LO1 List the characteristics of the uniform distribution.
LO2 Compute probabilities by using the uniform distribution.
LO3 List the characteristics of the normal probability distribution.
LO4 Convert a normal distribution to the standard normal distribution.
LO5 Find the probability that an observation on a normally distributed
random variable is between two values.
LO6 Find probabilities using the Empirical Rule.
LO7 Approximate the binomial distribution using the normal
distribution.
LO8 Describe the characteristics and compute probabilities using the
exponential distribution.
7-2
3. LO1 List the characteristics of
the uniform distribution.
The Uniform Distribution
The uniform probability
distribution is perhaps
the simplest
distribution for a
continuous random
variable.
This distribution is
rectangular in shape
and is defined by
minimum and maximum
values.
7-3
5. LO2 Compute probabilities by
using the uniform distribution.
The Uniform Distribution - Example
Southwest Arizona State University provides bus service to
students while they are on campus. A bus arrives at the North
Main Street and College Drive stop every 30 minutes between 6
A.M. and 11 P.M. during weekdays. Students arrive at the bus stop
at random times. The time that a student waits is uniformly
distributed from 0 to 30 minutes.
1. Draw a graph of this distribution.
2. Show that the area of this uniform distribution is 1.00.
3. How long will a student “typically” have to wait for a bus? In other
words what is the mean waiting time? What is the standard
deviation of the waiting times?
4. What is the probability a student will wait more than 25 minutes
5. What is the probability a student will wait between 10 and 20
minutes?
7-5
8. LO2
The Uniform Distribution - Example
3. How long will a student
“typically” have to wait for a
bus? In other words what is
the mean waiting time?
What is the standard
deviation of the waiting
times?
7-8
9. LO2
The Uniform Distribution - Example
4. What is the
P (25 Wait Time 30) (height)(b ase)
probability a
1
student will wait (5)
(30 0)
more than 25
minutes? 0.1667
7-9
10. LO2
The Uniform Distribution - Example
5. What is the
P (10 Wait Time 20) (height)(b ase)
probability a
1
student will wait (10 )
(30 0)
between 10 and 20
minutes? 0.3333
7-10
11. LO3 List the characteristics of the
normal probability distribution.
Characteristics of a Normal
Probability Distribution
1. It is bell-shaped and has a single peak at the center of the
distribution.
2. It is symmetrical about the mean
3. It is asymptotic: The curve gets closer and closer to the X-
axis but never actually touches it. To put it another way, the
tails of the curve extend indefinitely in both directions.
4. The location of a normal distribution is determined by the
mean, , the dispersion or spread of the distribution is
determined by the standard deviation,σ .
5. The arithmetic mean, median, and mode are equal
6. The total area under the curve is 1.00; half the area under
the normal curve is to the right of this center point, the mean,
and the other half to the left of it.
7-11
13. LO3
The Family of Normal Distribution
Equal Means and Different Different Means and
Standard Deviations Standard Deviations
Different Means and Equal Standard Deviations
7-13
14. LO4 Convert a normal distribution to the
standard normal distribution.
The Standard Normal Probability
Distribution
The standard normal distribution is a normal
distribution with a mean of 0 and a standard
deviation of 1.
It is also called the z distribution.
A z-value is the signed distance between a
selected value, designated X, and the population
mean , divided by the population standard
deviation, σ.
The formula is:
7-14
15. LO6 Find probabilities using the
Empirical Rule.
The Empirical Rule
About 68 percent of
the area under the
normal curve is within
one standard
deviation of the
mean.
About 95 percent is
within two standard
deviations of the
mean.
Practically all is
within three standard
deviations of the
mean.
7-15
16. LO6
The Empirical Rule - Example
As part of its quality
assurance program, the
Autolite Battery Company
conducts tests on battery
life. For a particular D-cell
alkaline battery, the mean
life is 19 hours. The useful
life of the battery follows a
normal distribution with a
standard deviation of 1.2
hours.
Answer the following questions.
1. About 68 percent of the
batteries failed between
what two values?
2. About 95 percent of the
batteries failed between
what two values?
3. Virtually all of the batteries
failed between what two
values?
7-16
18. LO5 Find the probability that an observation on a normally
distributed random variable is between two values.
The Normal Distribution – Example
The weekly incomes of
shift foremen in the
glass industry follow the
normal probability
distribution with a mean
of $1,000 and a
standard deviation of
$100.
What is the z value for
the income, let’s call it X,
of a foreman who earns
$1,100 per week? For a
foreman who earns
$900 per week?
7-18
19. LO5
Normal Distribution – Finding Probabilities
In an earlier example
we reported that the
mean weekly income
of a shift foreman in
the glass industry is
normally distributed
with a mean of $1,000
and a standard
deviation of $100.
What is the likelihood
of selecting a foreman
whose weekly income
is between $1,000
and $1,100?
7-19
21. LO5
Finding Areas for Z Using Excel
The Excel function
=NORMDIST(x,Mean,Standard_dev,Cumu)
=NORMDIST(1100,1000,100,true)
generates area (probability) from
Z=1 and below
7-21
22. LO5
Normal Distribution – Finding Probabilities
(Example 2)
Refer to the information
regarding the weekly income
of shift foremen in the glass
industry. The distribution of
weekly incomes follows the
normal probability
distribution with a mean of
$1,000 and a standard
deviation of $100.
What is the probability of
selecting a shift foreman in
the glass industry whose
income is:
Between $790 and $1,000?
7-22
23. LO5
Normal Distribution – Finding Probabilities
(Example 3)
Refer to the information
regarding the weekly income
of shift foremen in the glass
industry. The distribution of
weekly incomes follows the
normal probability
distribution with a mean of
$1,000 and a standard
deviation of $100.
What is the probability of
selecting a shift foreman in
the glass industry whose
income is:
Less than $790?
7-23
24. LO5
Normal Distribution – Finding Probabilities
(Example 4)
Refer to the information
regarding the weekly income
of shift foremen in the glass
industry. The distribution of
weekly incomes follows the
normal probability
distribution with a mean of
$1,000 and a standard
deviation of $100.
What is the probability of
selecting a shift foreman in
the glass industry whose
income is:
Between $840 and $1,200?
7-24
25. LO5
Normal Distribution – Finding Probabilities
(Example 5)
Refer to the information
regarding the weekly income
of shift foremen in the glass
industry. The distribution of
weekly incomes follows the
normal probability
distribution with a mean of
$1,000 and a standard
deviation of $100.
What is the probability of
selecting a shift foreman in
the glass industry whose
income is:
Between $1,150 and $1,250
7-25
26. LO5
Using Z in Finding X Given Area - Example
Layton Tire and Rubber
Company wishes to set a
minimum mileage guarantee on
its new MX100 tire. Tests
reveal the mean mileage is
67,900 with a standard
deviation of 2,050 miles and
that the distribution of miles
follows the normal probability
distribution. Layton wants to set
the minimum guaranteed
mileage so that no more than 4
percent of the tires will have to
be replaced.
What minimum guaranteed
mileage should Layton
announce?
26
7-26
27. LO5
Using Z in Finding X Given Area - Example
Solve X using the formula :
x- x 67,900
z
2,050
The value of z is found using the 4% information
The areabetween67,900 and x is 0.4600,found by 0.5000 - 0.0400
Using Appendix B.1, the area closest to 0.4600is 0.4599,which
gives a z alue of - 1.75. Then substituting into the equation :
x - 67,900
- 1.75 , then solving for x
2,050
- 1.75(2,050) x - 67,900
x 67,900 - 1.75(2,050)
x 64,312
7-27
29. LO7 Approximate the binomial distribution
using the normal distribution.
Normal Approximation to the Binomial
The normal distribution (a continuous distribution)
yields a good approximation of the binomial
distribution (a discrete distribution) for large values
of n.
The normal probability distribution is generally a good
approximation to the binomial probability distribution
when n and n(1- ) are both greater than 5.
7-29
30. LO7
Normal Approximation to the Binomial
Using the normal distribution (a continuous distribution) as a substitute
for a binomial distribution (a discrete distribution) for large values of n
seems reasonable because, as n increases, a binomial distribution gets
closer and closer to a normal distribution.
7-30
31. LO7
Continuity Correction Factor
The value .5 subtracted or added, depending on the
problem, to a selected value when a binomial probability
distribution (a discrete probability distribution) is being
approximated by a continuous probability distribution (the
normal distribution).
7-31
32. LO7
How to Apply the Correction Factor
Only one of four cases may arise:
1. For the probability at least X occurs, use the area above (X -.5).
2. For the probability that more than X occurs, use the area above
(X+.5).
3. For the probability that X or fewer occurs, use the area below (X -
.5).
4. For the probability that fewer than X occurs, use the area below
(X+.5).
7-32
33. LO7
Normal Approximation to the Binomial -
Example
Suppose the
management of the
Santoni Pizza Restaurant
found that 70 percent of
its new customers return
for another meal. For a
week in which 80 new
(first-time) customers
dined at Santoni’s, what
is the probability that
60 or more will return
for another meal?
7-33
34. LO7
Normal Approximation to the Binomial - Example
Binomial distribution solution:
P(X ≥ 60) = 0.063+0.048+ … + 0.001) = 0.197
7-34
35. LO7
Normal Approximation to the Binomial -
Example
Step 1. Find the
mean and the
variance of a binomial
distribution and find
the z corresponding
to an X of 59.5 (x-.5,
the correction factor)
Step 2: Determine
the area from 59.5
and beyond
7-35
36. LO8 Describe the characteristics and compute
probabilities using the exponential distribution.
The Family of Exponential Distributions
Characteristics and Uses:
1. Positively skewed, similar to
the Poisson distribution (for
discrete variables).
2. Not symmetric like the
uniform and normal
distributions.
3. Described by only one
parameter, which we identify
The exponential distribution usually describes
as λ, often referred to as the
inter-arrival situations such as:
“rate” of occurrence • The service times in a system.
parameter. • The time between “hits” on a web site.
4. As λ decreases, the shape of • The lifetime of an electrical component.
the distribution becomes “less • The time until the next phone call arrives in a
skewed.” customer service center
7-36
37. LO8
Exponential Distribution - Example
Orders for prescriptions arrive at a
pharmacy management website
according to an exponential
probability distribution at a mean of
one every twenty seconds.
Find the probability the next order
arrives in:
1) in less than 5 seconds,
2) in more than 40 seconds,
3) or between 5 and 40 seconds.
7-37
38. LO8
P ( Arrival 40) 1 P ( Arrival 40)
1
P ( Arrival 5) 20
( 40 )
1
1 (1 e )
(5)
1 (1 e 20
) 1 0.8647
1 0.7788 0.1353
0.2212
7-38
39. LO8
Exponential Distribution - Example
Compton Computers wishes to set a
minimum lifetime guarantee on it new
power supply unit. Quality testing
shows the time to failure follows an
exponential distribution with a mean of
4000 hours. Note that 4000 hours is a
mean and not a rate. Therefore, we
must compute λ as 1/4000 or 0.00025
failures per hour.
Compton wants a warranty period such
that only five percent of the power
supply units fail during that period.
What value should they set for the
warranty period?
7-39
40. LO8
Use formula (7–7) . In this case, the rate parameter is 4,000 hours and
we want the area, as shown in the diagram, to be .05.
P(Arrival Time x) 1 e( x)
1
(x)
4 , 000
0.05 1 e
Now, we need to solve this equation for x.
Obtain the natural log of both sides of the equation:
X = 205.17. Hence, Compton can set the warranty period at 205 hours
and expect about 5 percent of the power supply units to be returned.
7-40