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It includes various cases and practice problems related to Binomial, Poisson & Normal Distributions. Detailed information on where tp use which probability.
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Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution. Suppose we flip a coin two times and count the number of heads (successes).
It includes various cases and practice problems related to Binomial, Poisson & Normal Distributions. Detailed information on where tp use which probability.
this ppt gives you adequate information about Karl Pearsonscoefficient correlation and its calculation. its the widely used to calculate a relationship between two variables. The correlation shows a specific value of the degree of a linear relationship between the X and Y variables. it is also called as The Karl Pearson‘s product-moment correlation coefficient. the value of r is alwys lies between -1 to +1. + 0.1 shows Lower degree of +ve correlation, +0.8 shows Higher degree of +ve correlation.-0.1 shows Lower degree of -ve correlation. -0.8 shows Higher degree of -ve correlation.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Raimundo Soto - Catholic University of Chile
ERF Training on Advanced Panel Data Techniques Applied to Economic Modelling
29 -31 October, 2018
Cairo, Egypt
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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.
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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
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!
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2. Meaning and definitions
• Interpolation is the process or technique of finding the missing value within the given data.
Extrapolation is the process of finding the future value which lies outside the given set of data.
Definition:
According to W.M. Harper “interpolation consists in reading a value, which lies between two
extreme points. Extrapolation means reading that lies outside the two extreme points.”
3. Lets understand this concepts with examples
• Example 1:
• Example2: Find the sales value for 2021
X 1 2 3 4 5
Y 10 15 ------ 25 24
Year 2016 2017 2018 2019 2020
sales 100 120 130 125 145
4. The applications of the above techniques is based on the following
assumptions.
• 1. there are no sudden jumps, from one period to another period.
• 2. there is a uniformity in the rate of change in the values of variables.
• 3. there is a definite and stable relationships between the two variables.
5. Uses / significance of interpolation and extrapolation.
• The technique are helpful in filling the gaps that exist in statistical data.
• They are help full in obtaining the median and mode in continuous series.
• They are required when the data insufficient due to difficulties in collecting .
• They are use full in obtaining the future data also.
6. Methods of interpolation and extrapolation
• There are two methods of interpolation and extrapolation. They are
• Binominal Expansion method
• Newton’s method of advancing differences.
• Binominal Expansion method:
• This method is used for both technique and applicable only when following conditions are
fulfilled by the given or collected data.
• 1. The independent variables(x) advances by equal intervals like 2,4,6,8 or 3,6,9,12 and so on.
• 2. the value of (x), for which the value of (y) is to be interpolated, is one of the class limits of X
series.
7. There are two ways to adopt binominal expansion method.
1.By using Binominal Theorem.
2. BY using Pascal’s Triangle.
• steps to be followed in above method as follows.
Under Binominal Theorem Under Pascal’s triangle
Step 1. Find leading differences Find coefficients using Pascal’s triangle
Step2. Analysis of the Differences Formation of equations
Step3. finding the value. Find the values.
8. Illustration1.Solving the problem using Binominal Theorem when known values are Two.
X 1 2 3
Y 4 ---- 14
Leading difference
X Y ∆1
∆2
1 4 𝑦0 -4(𝑦1 − 𝑦0) = ∆0
1
2 -- 𝑦1 14(𝑦2 − 𝑦1)=∆1
1
18(∆1
1
−∆0
1
)
3 14 𝑦2
12. Using above steps we can able to get following formulas and same
formulas can apply directly for finding values.
13. Under Pascal’s triangle
• Find coefficients using Pascal’s triangle
• Formation of equations
• 1. Record the ‘Y’ alphabet with the
subscripts up to 𝑦0.
• 2. place the signs alternatively starting
from the first which must be Plus.
14. Illustration 1:Find the sales value for the year 2020 from the data given below.
year Sales in 00units
2014 240
2015 300
2016 320
2017 360
2018 400
2019 450
• Workings:
• Here 6years sales values are given hence
we needs to prepare 7rows pascal’s
triangle.
15. • After preparation of triangle, assign ‘Y’
alphabet with subscripts to known values.
year Sales in
2014 240(𝑦0)
2015 300(𝑦1)
2016 320((𝑦2)
2017 360(𝑦3)
2018 400(𝑦4)
2019 450(𝑦5)
2020 -------(𝑦6)
Place the signs alternatively starts from plus sign.
+(𝑦6)-(𝑦5)+ (𝑦4)- (𝑦3)+ (𝑦2)- (𝑦1)+ (𝑦0)
Join the last rows triangle value to above and
frame the formula.
(𝑦6)-(6𝑦5)+(15𝑦4)-(20𝑦3)+ (15𝑦2)- (6𝑦1)+ (𝑦0)=0
Using above formula find the needed value.
16. • (𝑦6)-(6𝑦5)+(15𝑦4)-(20𝑦3)+ (15𝑦2)- (6𝑦1)+ (𝑦0)=0
• (𝑦6)-(6*450)+(15*400)-(20*360)+ (15*320)- (6*300)+ (240)=0
• 𝑦6-2700+6000-7200+4800-1800+240=0
• 𝑦6-11700+11040=0
• 𝑦6-660=0(shift -660 to RHS)
• 𝑦6=660
• So sales value for 2020 is 66000 units.
17. Illustration 2:Find the sales value for the year 2017 from the data given
below.
year Sales in 00units
2014 150(𝑦0)
2015 235(𝑦1)
2016 365((𝑦2)
2017 ----(𝑦3)
2018 525(𝑦4)
2019 780(𝑦5)
• Workings:
• Here 5years sales values are given hence
we needs to prepare 6rows pascal’s
triangle.
18. • Place the signs alternatively starts from plus sign.
• (𝑦5)- (𝑦4)+ (𝑦3)- (𝑦2)+(𝑦1)- (𝑦0)
• Join the last rows triangle value to above and frame the formula.
• (𝑦5)- (5𝑦4)+ (10𝑦3)- (10𝑦2)+(5𝑦1)- (𝑦0)=0
Using above formula find the needed value.
(𝑦5)- (5𝑦4)+ (10𝑦3)- (10𝑦2)+(5𝑦1)- (𝑦0)=0
780- (5*525)+ (10𝑦2)- (10*365)+(5*235)- (150)=0
780-2625+(10𝑦2)-3650+1175-150=0
10𝑦2 +1955-6425=0
10𝑦2 -4470=0
10𝑦2 =4470
𝑦2 =4470/10
𝑦2 =447
19. Newton’s method of Advancing differences.
• This method is used only in interpolation, more concerned
with the interpolation of figures of (y) between two values of
‘X’ variables.it is based on advancing differences. We have to
proceed with the obtaining of differences between the ‘Y’
values extending the process till only one difference remains in
the end.
20. Following are the steps to be followed in applying Newton’s
formula
• Step1. find the leading differences, until only one differences remain in the end.
• step2. find the ‘X’ value by using the following formula
• X= item of interpolation -- item of origin
Difference between Adjoining items
Step3. Apply the Newton’s formula
Y= 𝑦0+X∆0
1
+x(x-1) ∆0
2
+x(x-1)(x-2) ∆0
3
+ x(x-1)(x-2)(x-3) ∆0
4
--------------so on
1x2 1x2x3 1x2x3x4
21. X Y
10 50
20 150
30 300
40 500
50 700
60 800
Illustration 1: Below are the wages earned by workers per month in a certain factory
interpolate the number of workers earning ₹35.
25. Illustration 2: Below are the wages earned by workers per month in a certain factory
interpolate the number of workers earning ₹25.
X Y
10 50
20 150
30 300
40 500
50 700
60 800