1) The document discusses discrete random variables through examples and calculations of expectation, variance, and probability.
2) It provides the probability distribution for a car share scheme and calculates the expectation of 2. It also gives an example of a discrete random variable and calculates its expectation, variance using two methods.
3) The final example gives Laura's probability distribution for her weekly milk bill and calculates the mean, standard deviation, and probabilities.
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Chapter 5: Discrete Probability Distribution
5.2 - Binomial Probability Distributions
Slides for a lecture by Todd Davies on "Probability", prepared as background material for the Minds and Machines course (SYMSYS 1/PSYCH 35/LINGUIST 35/PHIL 99) at Stanford University. From a video recorded July 30, 2019, as part of a series of lectures funded by a Vice Provost for Teaching and Learning Innovation and Implementation Grant to the Symbolic Systems Program at Stanford, with post-production work by Eva Wallack. Topics include Basic Probability Theory, Conditional Probability, Independence, Philosophical Foundations, Subjective Probability Elicitation, and Heuristics and Biases in Human Probability Judgment.
LECTURE VIDEO: https://youtu.be/tqLluc36oD8
EDITED AND ENHANCED TRANSCRIPT: https://ssrn.com/abstract=3649241
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Chapter 6: Normal Probability Distribution
6.1: The Standard Normal Distribution
1. continuous probability distribution
2. Normal Distribution
3. Application of Normal Dist
4. Characteristics of normal distribution
5.Standard Normal Distribution
Please Subscribe to this Channel for more solutions and lectures
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Chapter 5: Discrete Probability Distribution
5.2 - Binomial Probability Distributions
Slides for a lecture by Todd Davies on "Probability", prepared as background material for the Minds and Machines course (SYMSYS 1/PSYCH 35/LINGUIST 35/PHIL 99) at Stanford University. From a video recorded July 30, 2019, as part of a series of lectures funded by a Vice Provost for Teaching and Learning Innovation and Implementation Grant to the Symbolic Systems Program at Stanford, with post-production work by Eva Wallack. Topics include Basic Probability Theory, Conditional Probability, Independence, Philosophical Foundations, Subjective Probability Elicitation, and Heuristics and Biases in Human Probability Judgment.
LECTURE VIDEO: https://youtu.be/tqLluc36oD8
EDITED AND ENHANCED TRANSCRIPT: https://ssrn.com/abstract=3649241
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.1: The Standard Normal Distribution
1. continuous probability distribution
2. Normal Distribution
3. Application of Normal Dist
4. Characteristics of normal distribution
5.Standard Normal Distribution
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Statistics Assignment 1 HET551 – Design and Developm.docxrafaelaj1
Statistics Assignment 1
HET551 – Design and Development Project 1
Michael Allwright
Haddon O’Neill
Tuesday, 24 May 2011
1 Normal Approximation to the Binomial Distribution
This section of the assignment shows how a normal curve can be used to approximate the binomial distribution. This
section of the assignment was completed using a MATLAB function (shown in Listings 1) which would generate and
save plots of the various Binomial Distributions after normalisation, and then calculate the errors between the standard
normal curve and the binomial distribution.
The plots in Figures 1 and 2 show the binomial distribution for various n trials with probability p = 1
3
and p = 1
2
respectively. These binomial plots have been normalised so that they can be compared with the standard normal
distribution.
From these plots it can be seen that once the binomial distribution has been normalised, the normal approximation is
a good approach to estimating the binomial distribution. To determine its accuracy, the data in Table 1 shows the
evaluation of qn = P(bn ≥ µn + 2σn) for both the normal curve and binomial distribution.
qn = P(bn ≥ µn + 2σn) Calculation Error
n N(0, 1) B(n, 1
2
) B(n, 1
3
) B(n, 1
2
) B(n, 1
3
)
1 0.0228 0.0000 0.0000 -0.02278 -0.02278
2 0.0228 0.0000 0.0000 -0.02278 -0.02278
3 0.0228 0.0000 0.0370 -0.02278 0.01426
4 0.0228 0.0000 0.0123 -0.02278 -0.01043
5 0.0228 0.0313 0.0453 0.00847 0.02249
10 0.0228 0.0107 0.0197 -0.01203 -0.00312
20 0.0228 0.0207 0.0376 -0.00208 0.01486
30 0.0228 0.0214 0.0188 -0.00139 -0.00398
40 0.0228 0.0192 0.0214 -0.00354 -0.00134
50 0.0228 0.0164 0.0222 -0.00636 -0.00059
100 0.0228 0.0176 0.0276 -0.00518 0.00479
Table 1: Calculating the error of the normal approximation to the binomial for various n and p
2 Analytical investigation of the Exponential Distribution
For this part of the assignment the density function shown in Equation 1 was given.
f(x) = λe−λx for x ≥ 0 and λ ≥ 0 (1)
Before any calculations were attempted, the area under graph was checked to show that
´∞
−∞f(x) dx = 1. That is
that the total probability of all possible values was 1.
2.1 Derivation of CDF
To find the CDF of the given function, the function was integrated with 0 and x being the lower and upper bound
respectively. This derivation is shown in Equations 2 to 4.
CDF =
ˆ x
o
f(x) dx =
ˆ x
o
λe−λx dx (2)
2
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Number of Successes (shifted left by u = 0.33)
P
ro
b
a
b
ili
ty
(a) n = 1
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Number of Successes (shifted left by u = 0.67)
P
ro
b
a
b
ili
ty
(b) n = 2
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Number of Successes (shifted left by u = 1.00)
P
ro
b
a
b
ili
ty
(c) n = 3
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Number of Successes.
The aim of this presentation is to revise the functional regression models with scalar response (Linear, Nonlinear and Semilinear) and the extension to the more general case where the response belongs to the exponential family (binomial, poisson, gamma, ...). This extension allows to develop new functional classification methods based on this regression models. Some examples along with code implementation in R are provided during the talk. Lecturer: Manuel Febrero Bande, Univ. de Santiago de Compostela, Spain.
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 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.
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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.
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
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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.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
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Antifertility, Toxicity studies as per OECD guidelines
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This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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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.
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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Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
1. Statistics 1 Discrete Random Variables Section 2
1
DISCRETE RANDOM VARIABLES
Section 2
Choose from the following:
Introduction: Car share scheme a success
Example 4.3: A discrete random variable
Example 4.4: Laura’s Milk Bill
End presentation
2. Statistics 1 Discrete Random Variables Section 2
2
Car share scheme a success
Number of people /
Outcome r
1 2 3 4 5 > 5
Relative frequency /
Probability P(X = r)
0.35 0.375 0.205 0.065 0.005 0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 1 2 3 4 5 6
r
P(X = r )
3. Statistics 1 Discrete Random Variables Section 2
3
Expectation
r P(X = r) r P(X = r)
1 0.35 0.35
2 0.375
3 0.205
4 0.065
5 0.005
Totals 1
Multiply each r
value by P(X = r)
to form the
r P(X = r)
column
4. Statistics 1 Discrete Random Variables Section 2
4
Expectation
r P(X = r) r P(X = r)
1 0.35 0.35
2 0.375 0.75
3 0.205
4 0.065
5 0.005
Totals 1
Multiply each r
value by P(X = r)
to form the
r P(X = r)
column
5. Statistics 1 Discrete Random Variables Section 2
5
Expectation
r P(X = r) r P(X = r)
1 0.35 0.35
2 0.375 0.75
3 0.205 0.615
4 0.065
5 0.005
Totals 1
Multiply each r
value by P(X = r)
to form the
r P(X = r)
column
6. Statistics 1 Discrete Random Variables Section 2
6
Expectation
r P(X = r) r P(X = r)
1 0.35 0.35
2 0.375 0.75
3 0.205 0.615
4 0.065 0.26
5 0.005
Totals 1
Multiply each r
value by P(X = r)
to form the
r P(X = r)
column
7. Statistics 1 Discrete Random Variables Section 2
7
Expectation
r P(X = r) r P(X = r)
1 0.35 0.35
2 0.375 0.75
3 0.205 0.615
4 0.065 0.26
5 0.005 0.025
Totals 1
Multiply each r
value by P(X = r)
to form the
r P(X = r)
column
8. Statistics 1 Discrete Random Variables Section 2
8
Expectation
r P(X = r) r P(X = r)
1 0.35 0.35
2 0.375 0.75
3 0.205 0.615
4 0.065 0.26
5 0.005 0.025
Totals 1 2
Now find the total
of the
r P(X = r)
column
9. Statistics 1 Discrete Random Variables Section 2
9
Expectation
E(X) = m = S r P(X = r)
= 1×0.35 + 2×0.375 + 3×0.205 + 4×0.065 + 5×0.005
= 0.35 + 0.75 + 0.615 + 0.26 + 0.025
= 2
r P(X = r) r P(X = r)
1 0.35 0.35
2 0.375 0.75
3 0.205 0.615
4 0.065 0.26
5 0.005 0.025
Totals 1 2
Expectation
= E(X) or m
12. Statistics 1 Discrete Random Variables Section 2
12
Example 4.3
The discrete random variable X has the distribution:
(i) Find E(X).
(ii) Find E(X2).
(iii) Find Var(X) using
(a) E(X2) – m2 and (b) E([X – m]2) .
r 0 1 2 3
P(X = r) 0.2 0.3 0.4 0.1
13. Statistics 1 Discrete Random Variables Section 2
13
Example 4.3 : (i) Expectation E(X)
r P(X = r) r P(X = r)
0 0.2 0
1 0.3 0.3
2 0.4 0.8
3 0.1 0.3
Totals 1 1.4
Expectation
= E(X) or m
E(X) = m = S r P(X = r)
= 0×0.2 + 1×0.3 + 2×0.4 + 3×0.1
= 0 + 0.3 + 0.8 + 0.3
= 1.4
17. Statistics 1 Discrete Random Variables Section 2
17
Example 4.4 : Laura’s Milk Bill
Laura has one pint of milk on three days
out of every four and none on the fourth
day. A pint of milk costs 40p.
Let X represent her weekly milk bill.
(i) Find the probability distribution for her weekly milk
bill.
(ii) Find the mean (m) and standard deviation (s) of her
weekly milk bill.
(iii) Find (a) P(X > m + s ) and (b) P(X < m −s ).
18. Statistics 1 Discrete Random Variables Section 2
18
Example 4.4 : (i) Probability distribution
Since Laura has milk delivered, it takes four weeks
before the delivery pattern starts to repeat.
M Tu W Th F Sa Su No. pints Milk bill
6 £2.40
5 £2.00
5 £2.00
5 £2.00
r 2.00 2.40
P(X = r) 0.75 0.25
19. Statistics 1 Discrete Random Variables Section 2
19
Example 4.4 : (i) Mean μ or expectation E(X)
r P(X = r) r P(X = r)
2.00 0.75 1.50
2.40 0.25 0.60
Totals 1 2.10
Expectation
= E(X) or m
E(X) = m = S r P(X = r)
= 2.00 × 0.75 + 2.40 × 0.25
= 1.50 + 0.60
= 2.10
20. Statistics 1 Discrete Random Variables Section 2
20
Example 4.3 : (ii) Standard deviation σ
(a)
r P(X = r) r P(X = r) r2 P(X = r)
2.00 0.75 1.50 3.00
2.40 0.25 0.60 1.44
Totals 1 2.10 4.44
Var(X) = s 2 = S r 2 P(X = r) – m 2
= 22 × 0.75 + 2.42 × 0.25 – 2.12
= 3.00 + 1.44 – 2.12
= 4.44 – 4.41 = 0.03
Hence s = √0.03 = 0.17 (to 2 d.p.)
Using
Method A
21. Statistics 1 Discrete Random Variables Section 2
21
Example 4.4 : (iii) Calculating probabilities
r 2.00 2.40
P(X = r) 0.75 0.25
The probability distribution for Laura’s weekly milk bill:
(a) P(X > μ + σ) = P(X > 2.10 + 0.17)
= P(X > 2.27)
= 0.25
(b) P(X < μ − σ) = P(X < 2.10 − 0.17)
= P(X < 1.93)
= 0