This document discusses random variables and probability distributions. It begins by introducing random variables and how they can be either discrete or continuous. Discrete random variables can take on countable values, while continuous can be any value within a given interval. Several examples of each are provided like number of sales (discrete) and length (continuous). The document then discusses exploring random variables through an activity of tossing coins and calculating the number of tails. It also covers probability distributions for discrete random variables through graphs, tables, or formulas. Expected values and variance of discrete random variables are defined using summation notation.
Random Variable (Discrete and Continuous)Cess011697
Learning Competencies
- to recall statistical experiment and sample space
- to illustrate a random variable (discrete and continuous).
- to distinguish between a discrete and a continuous random variable.
Random Variable (Discrete and Continuous)Cess011697
Learning Competencies
- to recall statistical experiment and sample space
- to illustrate a random variable (discrete and continuous).
- to distinguish between a discrete and a continuous random variable.
Determining the Mean, Variance, and Standard Deviation of a Discrete Random Variable
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The Sum of Two Functions
The Difference of Two functions
The Product of Two Functions
The Quotient of Two Functions
The Product of A constant and a Function
Lesson in Introduction to Philosophy of Human Person
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: Random Variable, Discrete Random variable, Continuous random variable, Probability Distribution of Discrete Random variable, Mathematical Expectations and variance of a discrete random variable.
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about functions and its operations such as addition, subtraction, multiplication, division and composition. It is also comprised of some examples and exercises to be done for the said topic.
Determining the Mean, Variance, and Standard Deviation of a Discrete Random Variable
Visit the website for more services: https://cristinamontenegro92.wixsite.com/onevs
The Sum of Two Functions
The Difference of Two functions
The Product of Two Functions
The Quotient of Two Functions
The Product of A constant and a Function
Lesson in Introduction to Philosophy of Human Person
"Join me on my YouTube channel for more insightful topics! Don't forget to hit the subscribe button and share with your friends to stay updated on all the latest content!"
https://www.youtube.com/@JehnSimon
: Random Variable, Discrete Random variable, Continuous random variable, Probability Distribution of Discrete Random variable, Mathematical Expectations and variance of a discrete random variable.
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about functions and its operations such as addition, subtraction, multiplication, division and composition. It is also comprised of some examples and exercises to be done for the said topic.
Random variables and probability distributions Random Va.docxcatheryncouper
Random variables and probability distributions
Random Variable
The outcome of an experiment need not be a number, for example, the outcome when a
coin is tossed can be 'heads' or 'tails'. However, we often want to represent outcomes
as numbers. A random variable is a function that associates a unique numerical value
with every outcome of an experiment. The value of the random variable will vary from
trial to trial as the experiment is repeated.
There are two types of random variable - discrete and continuous.
A random variable has either an associated probability distribution (discrete random
variable) or probability density function (continuous random variable).
Examples
1. A coin is tossed ten times. The random variable X is the number of tails that are
noted. X can only take the values 0, 1, ..., 10, so X is a discrete random variable.
2. A light bulb is burned until it burns out. The random variable Y is its lifetime in
hours. Y can take any positive real value, so Y is a continuous random variable.
Expected Value
The expected value (or population mean) of a random variable indicates its average or
central value. It is a useful summary value (a number) of the variable's distribution.
Stating the expected value gives a general impression of the behaviour of some random
variable without giving full details of its probability distribution (if it is discrete) or its
probability density function (if it is continuous).
Two random variables with the same expected value can have very different
distributions. There are other useful descriptive measures which affect the shape of the
distribution, for example variance.
The expected value of a random variable X is symbolised by E(X) or µ.
If X is a discrete random variable with possible values x1, x2, x3, ..., xn, and p(xi)
denotes P(X = xi), then the expected value of X is defined by:
where the elements are summed over all values of the random variable X.
http://www.stats.gla.ac.uk/steps/glossary/probability_distributions.html#discvar
http://www.stats.gla.ac.uk/steps/glossary/probability_distributions.html#contvar
http://www.stats.gla.ac.uk/steps/glossary/probability_distributions.html#variance
If X is a continuous random variable with probability density function f(x), then the
expected value of X is defined by:
Example
Discrete case : When a die is thrown, each of the possible faces 1, 2, 3, 4, 5, 6 (the xi's)
has a probability of 1/6 (the p(xi)'s) of showing. The expected value of the face showing
is therefore:
µ = E(X) = (1 x 1/6) + (2 x 1/6) + (3 x 1/6) + (4 x 1/6) + (5 x 1/6) + (6 x 1/6) = 3.5
Notice that, in this case, E(X) is 3.5, which is not a possible value of X.
See also sample mean.
Variance
The (population) variance of a random variable is a non-negative number which gives
an idea of how widely spread the values of the random variable are likely to be; the
larger the variance, the more scattered the obser ...
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
2. In our previous study of mathematics, we
encountered the concept of probability. How do
we use this concept in making decisions concerning
a population using sample?
Decision- making is an important aspect in business,
education, insurance, and other real- life situations.
Many decisions are made by assigning probabilities
to all possible outcomes pertaining to the situation
and then evaluating the results.
4. OBJECTIVES:
At the end of this lesson, you are expected to:
➢ Illustrate a random variable
➢ Classify random variables as discrete or
continuous; and
➢ Find the possible values of a random variable.
5. Two Types of Random Variables
A random variable is a variable hat assumes
numerical values associated with the random
outcome of an experiment, where one (and only
one) numerical value is assigned to each sample
point.
6. Two Types of Random Variables
A discrete random variable can assume a countable
number of values.
▪ Number of steps to the top of the Eiffel Tower*
A continuous random variable can assume any value along
a given interval of a number line.
▪ The time a tourist stays at the top
once s/he gets there
7. Two Types of Random Variables
Discrete random variables
Number of sales
Number of calls
Shares of stock
People in line
Mistakes per page
Continuous random variables
Length
Depth
Volume
Time
Weight
8. Questions:
1.) How do you describe a discrete random variable?
2.) How do you describe a continuous random variable?
3.) Give three examples of discrete random variable.
4.) Give three examples of continuous random variable.
9. ACTIVITY:
TOSSING THREE COINS
Suppose three coins are tossed. Let Y be the random variable
representing the number of tails that occur. Find the values of the
random variable Y. Complete the table below.
POSSIBLE OUTCOMES VALUE OF THE RANDOM VARIABLE Y
(Number of Tails)
10. EXERCISE:
1.) Four coins are tossed. Let Z be the random variable representing the
number of heads that occur. Find the values of the random variable Z.
POSSIBLE OUTCOMES VALUE OF THE RANDOM VARIABLE Z
11. EXERCISE:
Let T be a random variable giving the number of heads plus the
number of tails in three tosses of a coin. List the elements of the
sample space S for the three tosses of the coin and assign a value to
each sample point.
POSSIBLE OUTCOMES VALUE OF THE RANDOM VARIABLE T
12. QUIZ 1
Classify the following random variables as discrete or continuous.
1.) the number of defective computers produced by manufacturer
2.) the weight of newborns each year in a hospital
3.) the number of siblings in a family of region
4.) the amount of paint utilized in a building project
5.) the number of dropout in a school district for a period of 10 years
6.) the speed of car
7.) the number of female athletes
8.) the time needed to finish the test
13. 9.) the amount of sugar in a cup of coffee
10.) the number of people who are playing LOTTO each day
11.) the number of accidents per year at an intersection
12.) the number of voters favoring a candidate
13.) the number of bushels of apples per hectare this year
14.) the number of patient arrivals per hour at medical clinic
15.) the average amount of electricity consumed per household
per month
14. Probability Distributions for Discrete
Random Variables
The probability distribution of a discrete random
variable is a graph, table or formula that specifies
the probability associated with each possible
outcome the random variable can assume.
p(x) ≥ 0 for all values of x
p(x) = 1
15. Probability Distributions
for Discrete Random
Variables
Say a random variable x follows
this pattern: p(x) = (.3)(.7)x-1
for x > 0.
This table gives the probabilities
(rounded to two digits) for x
between 1 and 10.
16. Expected Values of Discrete
Random Variables
The mean, or expected value, of a
discrete random variable is
( ) ( ).
E x xp x
= =
17. Expected Values of Discrete
Random Variables
The variance of a discrete random variable x is
The standard deviation of a discrete random variable x is
2 2 2
[( ) ] ( ) ( ).
E x x p x
= − = −
2 2 2
[( ) ] ( ) ( ).
E x x p x
= − = −
18. Expected Values of Discrete
Random Variables
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19. Expected Values of Discrete
Random Variables
In a roulette wheel in a U.S. casino, a $1 bet on
“even” wins $1 if the ball falls on an even
number (same for “odd,” or “red,” or “black”).
The odds of winning this bet are 47.37%
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