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Estadística para decisiones /Statistics for Decision Making
Universidad del Turabo
STAT 555 – DL- WORKSHOP FIVE
Topic: Continuous Probability Distribution
Introduction
In the previous workshops we dealt with distributions for discrete random variables,
which can only take certain values along an interval, with the possible values
having gaps between them. This workshop presents several continuous probability
distributions; these describe probabilities associated with random variables that are
able to assume any of an infinite number of values along an interval.
It’s important that everyone make their best effort to learn the normal distribution
because it will be the basis for understanding future topics for this course.
Specific Objectives:
At the end of the workshop, you will be able to:

Understand the nature and applications of the normal distribution.

Recognize the characteristics of the normal distribution.

Calculate probabilities using normal distribution.

Solve managerial situations using normal distribution.

Know the description and applications of exponential distribution.

Solve managerial situations using exponential distribution.
Language Objectives:

Analyze the situation, identify the problems and develop solutions for the
problems in English. Answer the assigned questions in English with the
correct language structure, syntax, and grammar.

Ask questions in a professional manner using strategic ideas and factors in
writing utilizing the correct terms in accordance with the course content and
using the proper rules in grammar.

Present statements and arguments showing leadership, professionalism and
considering the different opinions of the audience, in oral form with the
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proper pronunciation, using the correct grammar and verbs, ensuring the
terms and statements follow the content of the course.
Content
5.1
Continuous Probability Distribution
If a random variable is a continuous variable, its probability distribution is
called a continuous probability distribution. This type of probability
distribution differs from a discrete probability distribution in several ways:
 The probability that a continuous random variable will assume a
particular value is zero.
 As a result, a continuous probability distribution cannot be expressed
in tabular form.
 Instead, an equation or formula is used to describe a continuous
probability distribution.
These examples will clarify the difference between discrete and continuous
variables. Suppose the police department mandates that all policemen must
weigh between 150 and 250 pounds. The weight of a policeman would be an
example of a continuous variable; since a policeman’s weight could take on
any value between 150 and 250 pounds. Another example is when we flip a
coin and count the number of heads. The number of heads could be any
integer value between 0 and plus infinity. However, it could not be any
number between 0 and plus infinity. We could not, for example, get 2.5
heads. Therefore, the number of heads must be a discrete variable.
Continuous probability distributions have the following characteristics:
1. The vertical coordinate is a function of x, described as f(x) and
referred to as the probability density function.
2. The range of possible x values is along the horizontal axis.
3. The probability that x will take on a value between a and b will be the
area under the curve between points a and b, as shown in the next
figure. The probability density function, f(x), for a given continuous
distribution is expressed in algebraic terms, and the areas beneath are
obtained through the mathematics of calculus. However, tables are
provided in the text for readily identifying or calculating such areas.
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4. The total area under the curve will be equal to 1.0.
Area= P(a<x<b)
f(x)
a
b
Also, you are invited to watch the following presentation with audio about
this topic selecting the following link.
Continuous Probability Distribution:
http://www.youtube.com/watch?v=Fvi9A_tEmXQ
http://www.youtube.com/watch?v=szjL60gAweE
These videos present basic concepts of continuous probability distribution,
and examples of its application.
I invite you to practice and discover the wonders of continuous probability
distribution doing the next activity.
Activities
This activity does not have any points and will not be considered when assigning
the grade for the course, however, it will help you clarify any doubts and answer
any questions that you may have. Also, it will help you in getting prepared to do
the assignments and to take the short test. These will be part of the final evaluation
and you will find them at the end of the workshop.
Activity 5.1 Problems related to continuous probability distribution
The following problems are related to continuous probability distribution. Once you
consider that you understand these concepts, solve the problems.
1. What is the difference between a continuous probability distribution and a
discrete probability distribution?
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2. Why is the total area beneath a probability density function equal to 1.0?
3. Remember, you must submit your answers using Excel, Word or another
compatible program and save it with this name: W5.A1.name.lastname.
4. Finally, you are required to send your responses to the facilitator using the
Tareas/Tasks option on the menu, select Workshop Five/Practice 5.A1
and Click to launch for submit.
5. After the facilitator receives your response, it will be analyzed and you will
receive recommendations and comments (if any).
5.2
Normal Distribution
The most important probability distribution for describing a continuous
random variable is the normal probability distribution. The normal
distribution has been used in a wide variety of practical applications in which
the random variables are heights and weights of people, test-scores, etc.
Characteristics of the normal distribution:
1. The entire family of normal distribution is differentiated by two
parameters: the mean and the standard deviation.
2. The highest point on the normal curve is a mean, which is also the
median and mode of the distribution.
3. The mean of the distribution can be any numerical value: negative,
zero, or positive.
4. The normal distribution is symmetric, with the shape of the normal
curve (Bell-Shape). The tails of the normal curve extend to infinity in
both directions and theoretically never touch the horizontal axis. The
normal distribution is not skewed.
5. The standard deviation determines how flat and wide the normal curve
is. Larger values of the standard deviation result in wider, flatter
curves, showing more variability in the data.
6. Areas under the normal curve give probabilities for the normal random
variable. The total area under the curve for normal distribution is 1.
The area under the curve to the left of the mean is 0.50 and the area
under the curve to the right of the mean is 0.50.
7. The percentage of values in some commonly used variables are:
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a. 63.8% of the values of a normal random variable are within plus
or minus one standard deviation of its mean.
b. 95.4% of the values of a normal random variable are within plus
or minus two standard deviation of its mean.
c. 99.7% of the values of a normal random variable are within plus
or minus three standard deviation of its mean.
Normal curve
The form or shape is illustrated by the bell-shaped normal curve.
The z-score for a standard normal distribution:
z
x

where, z  the distance from the mean, measured in standard deviation units
x  the value of x in which w e are interested
  the mean of the distributi on
  the standard deviation of the distributi on
Along with this formula it is necessary to use the standard normal
distribution table.
To use the standard normal distribution table select the following link:
STAT555DL_ W5_5_2 Standard Normal Probability Table (MS
Word Document)
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Estadística para decisiones /Statistics for Decision Making
To resolve normal distribution problems with Microsoft Office (Excel) you
have to select in the main menu option ”Formulas”, then “Insert Function”.
This opens a dialog box where you will find the normal distribution option as
“NORMDIST”, once you check the box; the following box will be displayed:
Also, you are invited to watch the following video with audio about this
topic selecting the following link.
Normal Distribution
http://www.youtube.com/watch?v=sM-4jYNr90Y
Here an introduction to normal distribution is developed through a step by
step example.
Here you will find a new activity to practice normal distribution. Enjoy it.
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Activity 5.2 Problems related to normal distribution
The following problems are related to normal distribution. Once you consider that
you understand these concepts, solve the problems.
1. In 1999, the average charge for tax preparation by H&R Block, Inc. was
$84.57. Assuming a normal distribution and a standard deviation of   $10 ,
what proportion of H&R Block’s tax preparation fees were:
a. more than $84.57?
b. between $64.57 and $104.57?
c. between $74.57 and $94.57?
d. more than $104.57?
2. It has been reported that the average hotel check-in time, from curbside to
delivery of bags into the room, is 12.1 minutes. An Li has just left the cab
that brought her to her hotel. Assuming a normal distribution with a
standard deviation of 2.0 minutes, what is the probability that the time
required for An Li and her bags to get to the room will be:
a. greater than 14.1 minutes?
b. between 10.1 and 14.1 minutes?
c. less than 8.1 minutes?
d. between 10.1 and 16.1 minutes?
3. Remember, you must submit your answers using Excel, Word or another
compatible program and save it with this name: W5.A5.2.name.lastname
4. Finally, you are required to send your responses to the facilitator using the
Tareas/Tasks option on the menu, select Workshop Five/Practice 5.A2
and Click to launch for submit.
5. After the facilitator receives your response, it will be analyzed and you will
receive recommendations and comments (if any).
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5.3 Exponential Distribution
Exponential distributions are a type of continuous probability distribution. They
describe the times between events in a Poisson process. It is a process in
which events occur continuously and independently at a constant average rate.
Examples of variables that are approximately exponentially distributed are:

the time until you have your next car accident

the time until you get your next phone call
P ( x )    x
where,   the mathematic al constant, 2.71828
the base of the natural logarithm system
  the mean and variance of a Poisson distributi on
1  the mean and standard deviation of the

correspond ing exponentia l distributi on
P ( X  x )    x
P ( X  x )  1    x
P ( x1  X  x2 )  P ( X  x2 )  P ( X  x1 )    x1    x2
To resolve exponential distribution problems with Microsoft Office (Excel) you
have to select in the main menu option ”Formulas”, then “Insert Function”.
This opens a dialog box where you will find the normal distribution option as
“EXPONDIST”, once you check the box, the following box will be displayed:
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Also, you are invited to watch the following videos with audio about this
topic selecting the following links.
Exponential Distribution:
http://www.youtube.com/watch?v=crG5VlpS-RE
http://www.youtube.com/watch?v=E19gxHLpurc
http://www.youtube.com/watch?v=NSpBjtrIKqM
Through these presentations, you will have the opportunity to see
examples solved using exponential distribution. In addition, you can see
how to solve problems using Excel.
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Activity 5.3 Problems related to exponential distribution
Concentrate and tell me, what are the results of the following problem?
The following problems are related to exponential distribution. Once you consider
that you understand these concepts, solve the problems.
1. A random variable is Poisson distributed with   1.5 occurrences per hour.
For the corresponding exponential distribution, and x= hours until the next
occurrence, identify the mean of x and determine the following:
a. P(x>0.5)
b. P(x>1.0)
c. P(x>1.5)
d. P(x>2.0)
2. The owner of a self-service carwash has found that customers take an
average of 8 minutes to wash and dry their cars. Assuming that the time in
the self-service tends to be exponentially distributed, what is the probability
that a customer will require more than 10 minutes to complete the job?
3. Remember, you must submit your answers using Excel, Word or another
compatible program and save it with this name: W5.A5.3.name.lastname.
4. Finally, you are required to send your responses to the facilitator using the
Tareas/Tasks option on the menu, select Workshop Five/Practice 5.A3
and Click to launch for submit.
5. After the facilitator receives your response, it will be analyzed and you will
receive recommendations and comments (if any).
I invite you to participate in the following Forum where you can share any
thoughts or doubts you have about the topic in discussion.
To join us, access Comunicación/Communication link and press Workshop
Five: Doubts and Questions.
The assignments below will help you reflect on the discussed topic. Follow the
instructions provided for each assignment.
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Tasks/Activities:
Task 5.1 Part A Project (written)
Instructions
1. On the Internet, find and record observations on at least two different
continuous random variables such as stock market indices, interest rates,
corporate sales, or any business-related topic of interest to you.
2. Select one data and estimate the mean and standard deviation.
3. Next identify three events of interest and compute their probability.
4. Write two paragraphs using Word or a compatible program to discuss what
you have learned about normal distribution.
1. Save your file as: T5.1.name.lastname
5. The facilitator will use Appendix N: Workshop 5 – T5.1 Project
Evaluation Rubric (15 points) for this task. (See in the syllabus).
6. Submit to the facilitator using the link Tareas/Tasks selecting Workshop
Five/T5.1: Part A Project.
------------------------------------------------------------------------------------Task 5.1 Part B Project Voice Forum (oral)
Instructions
In addition, participate in a voice discussion forum T5.1, Part B: Project.
1. You can review the instructions or tutorial for recording accessing e-lab.
2. The facilitator will use the Appendix N- Workshop 5: T5.1 Part B-Project
Discussion Forum Rubric (10 points).
3. The duration of the recording should fluctuate between 1 and 3 minutes.
4. Please record your message through Tareas/Tasks, select Workshop Five
and press T5.1. Part B Project Voice forum link to say to your colleagues
what you have learned about normal distribution.
5. Read several of your classmate forums and react to at least two of them.
Points
The project will have a value of 20 points. (T5.1 Part A: 15; T5.1 Part B 10 points)
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Task 5.2 Short Test 5
Instructions
1. At the end of Workshop Five, you will take a short test. You should have
studied the topics in the workshop and answered the practice problems
before taking this test.
2. This test has ten (10) multiple-choice questions. To access the test, go to
the Tareas/Tasks option and select Workshop Five/Short Test 5–
Introduction to Statistics. You will see the page with the questions and
multiple choices. Every time you answer a question, it is important to press
“save” and when you finish the test, you must press “submit”.
3. Read each question carefully; do not answer the question until you are sure
what is being asked. You will see the points received, and any incorrect
answers you had (if any) with the correct answer.
Points
The short test will have a value of 20 points; each question is worth 2 points.
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Task 5.3 Voice Forum Reflective Journal (oral)
1. At the end of Workshop Five, you will prepare an oral work reflection
2. In this activity, you should answer the following questions:
a. Which topic did I learn?
b. How can I use it in my daily life?
c. What would you have added to the workshop?
d. Which topic impressed me the most? Why?
e. Which topic did I like the least? Why?
f. Which topic did I not understand?
3. You can review the instructions or tutorial for recording in the voice forum
accessing e-lab.
4. The facilitator will use Appendix P- Workshop Five: Reflective Journal to
evaluate this task and you can find it in the syllabus.
5. The duration of the recording should fluctuate between 1 and 3 minutes.
6. Please record your message through Tareas/Tasks and click Workshop
Five; press T5.3 Voice Forum: Reflective Journal.
Points
The work reflection is worth 10 points.
You have completed Week/Workshop Five!
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