JEE Mathematics/ Lakshmikanta Satapathy/ Theory of Probability part 9 which explains Random variables , its probability distribution, Mean of a random variable and Variance of a random variable
JEE Mathematics/ Lakshmikanta Satapathy/ Theory of probability part 10/ Bernoulli trials and Binomial distribution of probability of Bernoulli trials and probability function with example
JEE Mathematics/ Lakshmikanta Satapathy/ Questions and answers part 7 involving probability distribution and determination of mean and variance of a random variable
JEE Mathematics/ Lakshmikanta Satapathy/ Theory of probability part 10/ Bernoulli trials and Binomial distribution of probability of Bernoulli trials and probability function with example
JEE Mathematics/ Lakshmikanta Satapathy/ Questions and answers part 7 involving probability distribution and determination of mean and variance of a random variable
Mayo Slides: Part I Meeting #2 (Phil 6334/Econ 6614)jemille6
Slides Meeting #2 (Phil 6334/Econ 6614: Current Debates on Statistical Inference and Modeling (D. Mayo and A. Spanos)
Part I: Bernoulli trials: Plane Jane Version
Discrete Random Variable (Probability Distribution)LeslyAlingay
This presentation the statistics teachers to discuss discrete random variable since it includes examples and solutions.
Content:
-definition of random variable
-creating a frequency distribution table
- creating a histogram
-solving for the mean, variance and standard deviation.
References:
http://www.elcamino.edu/faculty/klaureano/documents/math%20150/chapternotes/chapter6.sullivan.pdf
https://www.mathsisfun.com/data/random-variables-mean-variance.html
https://www.youtube.com/watch?v=OvTEhNL96v0
https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch12/5214891-eng.htm
Detail Description about Probability Distribution for Dummies. The contents are about random variables, its types(Discrete and Continuous) , it's distribution (Discrete probability distribution and probability density function), Expected value, Binomial, Poisson and Normal Distribution usage and solved example for each topic.
JEE Mathematics/ Lakshmikanta Satapathy/ Indefinite Integration part 18/ Integration by parts 5/ Method involving product of exponential function with sum of two functions
Mayo Slides: Part I Meeting #2 (Phil 6334/Econ 6614)jemille6
Slides Meeting #2 (Phil 6334/Econ 6614: Current Debates on Statistical Inference and Modeling (D. Mayo and A. Spanos)
Part I: Bernoulli trials: Plane Jane Version
Discrete Random Variable (Probability Distribution)LeslyAlingay
This presentation the statistics teachers to discuss discrete random variable since it includes examples and solutions.
Content:
-definition of random variable
-creating a frequency distribution table
- creating a histogram
-solving for the mean, variance and standard deviation.
References:
http://www.elcamino.edu/faculty/klaureano/documents/math%20150/chapternotes/chapter6.sullivan.pdf
https://www.mathsisfun.com/data/random-variables-mean-variance.html
https://www.youtube.com/watch?v=OvTEhNL96v0
https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch12/5214891-eng.htm
Detail Description about Probability Distribution for Dummies. The contents are about random variables, its types(Discrete and Continuous) , it's distribution (Discrete probability distribution and probability density function), Expected value, Binomial, Poisson and Normal Distribution usage and solved example for each topic.
JEE Mathematics/ Lakshmikanta Satapathy/ Indefinite Integration part 18/ Integration by parts 5/ Method involving product of exponential function with sum of two functions
15 Probability Distribution Practical (HSC).pdfvedantsk1
Understanding Murphy's Law: Embracing the Unexpected
Content
Section 1: Unveiling Murphy's Law
Section 2: Real-life Applications
Section 3: Navigating the Unexpected
Section 1: Unveiling Murphy's Law
Page 1.1: Origin and Concept
Historical Context: Murphy's Law, originating from aerospace engineering, embodies the principle that "anything that can go wrong will go wrong." Its evolution from an engineering adage to a universal concept reflects its enduring relevance in diverse scenarios, providing a unique perspective on risk assessment and preparedness.
Psychological Implications: Understanding the law's impact on human behavior and decision-making processes provides insights into risk assessment, preparedness, and the psychology of uncertainty, offering valuable lessons for educators in managing unexpected events in the classroom.
Cultural Permeation: The law's integration into popular culture and its influence on societal perspectives toward unpredictability and risk management underscores its significance in contemporary discourse, highlighting its relevance in educational settings.
Page 1.2: The Science Behind the Law
Entropy and Probability: Exploring the scientific underpinnings of Murphy's Law reveals its alignment with principles of entropy and the probabilistic nature of complex systems, shedding light on its broader applicability, including its relevance in educational systems and institutional frameworks.
Complex Systems Theory: The law's resonance with the behavior of complex systems, including technological, social, and natural systems, underscores its relevance in diverse domains, from engineering to project management, offering insights into managing the complexities of educational environments.
Adaptive Strategies: Analysis of the law's implications for adaptive strategies and resilience planning offers valuable insights into mitigating the impact of unexpected events and enhancing system robustness, providing practical guidance for educators in navigating unforeseen challenges.
Page 1.3: Psychological and Behavioral Aspects
Cognitive Biases and Decision Making: Understanding how cognitive biases influence responses to unexpected events provides a framework for addressing the psychological dimensions of Murphy's Law in professional and personal contexts, offering strategies for educators to support students in managing unexpected outcomes.
Stress and Coping Mechanisms: Exploring the psychological impact of unexpected outcomes and the development of effective coping mechanisms equips individuals and organizations with strategies for managing uncertainty, providing valuable insights for educators in supporting students' emotional well-being.
Learning from Failure: Embracing the lessons inherent in Murphy's Law fosters a culture of learning from failure, promoting resilience, innovation, and adaptability in the face of unforeseen challenges, offering educators a framework for cultivating a growth mindset in students.
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces.
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the
conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
Common Fixed Point Theorems in Compatible Mappings of Type (P*) of Generalize...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings
under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy
metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
The monotone likelihood ratio test is the important part of statistics.
For your more information you can contact
Md.Sohel Rana
Jahangirnagar University
This presentation contains the topic as follows, probability distribution, random variable, continuous variable, discrete variable, probability mass function, expected value and variance and examples
JEE Physics/ Lakshmikanta Satapathy/ Work Energy and Power/ Force and Potential energy/ Angular momentum and Speed of Particle/ MCQ one or more correct
JEE Physics/ Lakshmikanta Satapathy/ MCQ On Work Energy Power/ Work-Energy theorem/ Work done by Gravity/ Work done by Air resistance/ Change in Kinetic Energy of body
CBSE Physics/ Lakshmikanta Satapathy/ Electromagnetism QA/ Magnetic field due to circular coil at center & on the axis/ Magnetic field due to Straight conductor/ Magnetic Lorentz force
CBSE Physics/ Lakshmikanta Satapathy/ Amplitudes of Reflected and Transmitted waves/ Sound as Pressure wave/ Speed of sound in Fluids/ Intensity and Loudness of sound
CBSE Physics/ Lakshmikanta Satapathy/ Wave motion/ Vibration of air columns/ Open & closed pipes/ Fundamental frequency & overtones/ End correction/ Resonance tube
CBSE Physics/ Lakshmikanta Satapathy/ Wave Motion Theory/ Reflection of waves/ Traveling and stationary waves/ Nodes and anti-nodes/ Stationary waves in strings/ Laws of transverse vibration of stretched strings
CBSE Physics/ Lakshmikanta Satapathy/ Wave theory/ path difference and Phase difference/ Speed of sound in a gas/ Intensity of wave/ Superposition of waves/ Interference of waves
JEE Mathematics/ Lakshmikanta Satapathy/ Definite integrals part 8/ JEE question on definite integral involving integration by parts solved with complete explanation
JEE Physics/ Lakshmikanta Satapathy/ Question on the magnitude and direction of the resultant of two displacement vectors asked by a student solved in the slides
JEE Mathematics/ Lakshmikanta Satapathy/ Quadratic Equation part 2/ Question on properties of the roots of a quadratic equation solved with the related concepts
JEE Mathematics/ Lakshmikanta Satapathy/ Probability QA part 12/ JEE Question on Probability involving the complex cube roots of unity is solved with the related concepts
JEE Mathematics/ Lakshmikanta Satapathy/ Inverse trigonometry QA part 6/ Questions on Inverse trigonometric functions involving tan inverse function solved with the related concepts
JEE Mathematics/ Lakshmikanta Satapathy/ Inverse Trigonometry QA part 5/ Question on sin inverse cosine inverse and tan inverse solved with the related concepts
JEE Physics/ Lakshmikanta Satapathy/ Transient current QA part 1/ JEE question on maximum and minimum current from a DC source connected across Inductance and Resistance solved with the related concepts
JEE Physics/ Lakshmikanta Satapathy/ Electromagnetism QA part 7/ Question on doubling the range of an ammeter by shunting solved with the related concepts
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
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!
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
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
2. Physics Helpline
L K Satapathy
Random Variable : In a random experiment, a single real number may be
assigned to each outcome () of the experiment ( S) , which may
be different for different outcomes.
Definition: A random variable is a real valued function whose domain is
the sample space of a random experiment.
For tossing two coins S = { HH , HT , TH , TT }
Let X denote ‘the number of heads’ obtained. Then X is a random
variable whose value for the different outcomes are as follows:
X(HH) = 2 , X(HT) = 1 , X(TH) = 1 and X(TT) = 0
We can define more than one random variable on the same sample space.
Let Y denote ‘the number of heads – the number of tails’ obtained.
Then Y(HH) = 2 – 0 =2 , Y(HT) = 1 – 1 = 0 ,
Y(TH) = 1 – 1 = 0 and Y(TT) = 0 – 2 = – 2
Probability Theory 9
3. Physics Helpline
L K Satapathy
Example: A person plays the game of tossing three coins. For each
head he is given Rs 3 by the organiser and for each tail , he has to give
Rs 2 to the organiser. Let X denote the amount gained by the person.
Show that X is a random variable and exhibit it as a function on the
sample space of the experiment.
Ans : For 3 coins, S = { HHH , HHT , HTH , HTT , THH , THT , TTH , TTT }
X = Amount gained = (3 number of heads) – (2 number of tails)
X (HHH) = 33 – 20 = 9 , X(HHT) = X(HTH) = X(THH) = 32 – 21 = 4
X(HTT) = X(THT) = X(TTH) = 31 – 22 = – 1 , X(TTT) = 30 – 23 = – 6
X is a real number whose value is defined for each outcome
and X takes a unique value for each outcome of the experiment
X is a function whose domain = S and whose range = { – 6 , – 1 , 4 , 9 }
Probability Theory 9
4. Physics Helpline
L K Satapathy
Probability distribution of a Random Variable :
A description of the values of a random variable X along with the corresponding
probabilities is called the probability distribution of the random variable X.
Consider the experiment of selecting 1 family out of 10 families 1 2 10, , . . . ,f f f
in such a manner that each family is equally likely to be selected. Let the families
have 3 , 4 , 3 , 2 , 5 , 4 , 3 , 6 , 4 , 5 members respectively.1 2 10, , . . . ,f f f
Let us select a family at random and note the number of members denoted by X .
X is a random variable defined as follows:
1 2 3 4 5( ) 3 , ( ) 4 , ( ) 3 , ( ) 2 , ( ) 5X f X f X f X f X f
6 7 8 9 10( ) 4 , ( ) 3 , ( ) 6 , ( ) 4 , ( ) 5X f X f X f X f X f
X = 2 for ( ) , X = 3 for ( ) , X = 4 for ( )4f 1 3 7, ,f f f
X = 5 for ( ) and X = 6 for ( )
2 6 9, ,f f f
5 10,f f 8f
Probability Theory 9
5. Physics Helpline
L K Satapathy
Each family is equally likely to be selected .
Probability of selecting each family (from a total of 10) = 1/10
X = 2 for 1 family P(X = 2) = 1/10
X = 3 for 3 families P(X = 3) = 3/10
X = 4 for 3 families P(X = 4) = 3/10
X = 5 for 2 families P(X = 5) = 2/10
X = 6 for 1 family P(X = 6) = 1/10
X 2 3 4 5 6
P(X) 1/10 3/10 3/10 2/10 1/10
Probability distribution
We observe that
1 3 3 2 1 10 1
10 10 10 10 10 10
Probability Theory 9
6. Physics Helpline
L K Satapathy
In general , if be the possible values of the random variable X1 2, , . . . , nx x x
X
P(X)
Probability distribution
1x 2x 3x nx
1p 2p 3p np
1
0, 1,2,3,..,
1
i
n
i
i
p i n
and p
Where
and the probability of X taking the value be ,( )i iP X x p ix
then the probability distribution of X is described as follows:
Probability Theory 9
7. Physics Helpline
L K Satapathy
Mean of a Random Variable :
It is the measure of the central tendency or the average value of a random variable.
Definition :
Let X be a random variable whose possible values 1 2, , . . . , nx x x
1 1 2 2
1
( ) . . . .
n
i i n n
i
E X x p x p x p x p
Then mean of X , denoted by is the weighted average of the possible values of X ,
each value being weighted by its probability of occurrence.
It is also called the expectation of X , denoted by E(X).
In other words , the mean or expectation of a random variable X is the sum of the
products of all the possible values of X with their respective probabilities.
occur with probabilities respectively.1 2, , . . . , np p p
Then
Probability Theory 9
8. Physics Helpline
L K Satapathy
Example : Let a pair of dice be thrown and the random variable X be the sum of the
numbers appearing on the two dice. Find the mean or expectation of X .
Answer : The sample space for throwing of two dice is
The random variable X ( sum of the numbers on the two dice ) can take any one of the
values 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 .
S consists of 36 elementary events which are in the form of ordered pairs (x , y) ,
where each of x and y can take the values of 1 , 2 , 3 , 4 , 5 or 6 .
Probability of each elementary event = 1/36 .
(1,1) , (1,2) , (1,3) , (1,4) , (1,5) , (1,6) ,
(2,1) , (2,2) , (2,3) , (2,4) , (2,5) , (2,6) ,
(3,1) , (3,2) , (3,3) , (3,4) , (3,5) , (3,6) ,
(4,1) , (4,2) , (4,3) , (4,4) , (4,5) , (4,6) ,
(5,1) , (5,2) , (5,3) , (5,4) , (5,5) , (5,6) ,
(6,1) , (6,2) , (6,3) , (6,4) , (6,5) , (6,6)
S =
Probability Theory 9
9. Physics Helpline
L K Satapathy
5( 8) {(2,6),(3,5),(4,4),(5,3),(6,2)}
36
P X P
4( 9) {(3,6),(4,5),(5,4),(6,3)}
36
P X P
3( 10) {(4,6),(5,5),(6,4)}
36
P X P
2( 11) {(5,6),(6,5)}
36
P X P
1( 12) {(6,6)}
36
P X P
1( 2) {(1,1)}
36
P X P Now
2( 3) {(1,2),(2,1)}
36
P X P
3( 4) {(1,3),(2,2),(3,1)}
36
P X P
4( 5) {(1,4),(2,3),(3,2),(4,1)}
36
P X P
5( 6) {(1,5),(2,4),(3,3),(4,2),(5,1)}
36
P X P
6( 7) {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}
36
P X P
Probability Theory 9
11. Physics Helpline
L K Satapathy
Variance of a Random Variable :
It is the variability or spread in the values of a random variable.
Definition :
probabilities respectively.1 2( ), ( ), . . . , ( )np x p x p x
Let X be a random variable whose possible values occur with1 2, , . . . , nx x x
Let be the mean of X . Then the variance of X is defined as( )E X
2 2
1
( ) ( ) . ( )
n
x i i
i
Var X x p x
Standard Deviation of a Random Variable :
It is the non-negative square root of the variance of a random variable defined as
2
1
( ) ( ) . ( )
n
x i i
i
Var X x p x
Probability Theory 9
12. Physics Helpline
L K Satapathy
Alternative expression for Variance of a Random Variable :
2 2
1
( ) ( ) . ( )
n
x i i
i
Var X x p x
2 2
1
( 2 ). ( )
n
i i i
i
x x p x
2 2
1 1 1
. ( ) . ( ) 2 . ( )
n n n
i i i i i
i i i
x p x p x x p x
2 2
1 1 1
. ( ) . ( ) 2 . . ( )
n n n
i i i i i
i i i
x p x p x x p x
2 2 2
1
. ( ) 2
n
i i
i
x p x
1 1
[ ( ) 1 . ( ) ]
n n
i i i
i i
Since p x and x p x
2 2
1
. ( )
n
i i
i
x p x
2 2 2
( ) ( ) [ ( )]xor Var X E X E X 2 2
1
[ , ( ) . ( ) ]
n
i i
i
where E X x p x
Probability Theory 9
13. Physics Helpline
L K Satapathy
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