See https://docs.google.com/presentation/d/1E0i1tI0gzUbxS1BNxaoCdVlu5ZpZ8A_D/edit#slide=id.p1

You need to start an instance and open the required port on Google Cloud. For detailed instructions, refer to this video. Additionally, ensure you increase the GPU quota to 1, as the default GPU quota on Google Cloud is set to 0.

I would recommend using the disk image Deep Learning on Linux -> Deep Learning VM with CUDA 12.4 M126.

Configure Jupyter notebook

There is a known issue with Jupyter Server under Anaconda. The easiest workaround is to downgrade jupyter-server as below.

pip install jupyter-server==1.24.0 

We also need to generate a jupyter notebook configuration file for further configuration, run

jupyter notebook --generate-config

Create jupyter password

jupyter notebook password

Run Jupyter

jupyter notebook --ip=0.0.0.0 --no-browser

The default port is 8888. You should now be able to connect to the Jupyter Notebook locally. Assuming the external IP is 33.11.2.11, open your browser and navigate to:

http://33.11.2.11:8888

Enter the password you selected, and you can start creating a new notebook.

Make life easier

In .jupyter/jupyter_notebook_config.py, uncomment and change the following

• c.NotebookApp.ip = ‘0.0.0.0’
• c.NotebookApp.open_browser = False
• c.NotebookApp.port = 8888 # or something else if you want to use a different port. Remember to open the port through the firewall setting if you do so.

Then you can run jupyter notebook simply by

jupyter notebook

Update:

GPUs are often unavailable during weekdays. In such cases, you may want to provision a spot instance instead of a standard instance. Spot instances are slightly cheaper but may terminate unexpectedly, typically with a brief warning.

To allocate a spot instance, go to the instance creation page. In the left-hand panel, select Advanced, then locate the Provisioning model option and choose SPOT. Keep in mind that this choice is permanent, and you cannot switch back to a standard instance once selected. A possible strategy is to create both a standard and a spot instance, using the spot instance when the standard instance is unavailable.

SUMMARY

The speaker discusses how maladaptive perfectionism hinders time management for PhD students and provides strategies to overcome it.

IDEAS:

• Perfectionism builds self-esteem but can hinder productivity and time management for PhD students.
• Maladaptive perfectionism creates unnecessary rules that restrict daily productivity and progress.
• Responding immediately to emails can waste time; it’s better to batch email responses.
• Prioritizing one or two significant tasks daily can reduce overwhelm and improve focus.
• Energy levels fluctuate throughout the day, affecting when to tackle challenging tasks.
• Saying no to non-prioritized tasks can free up time for essential responsibilities.
• Batching similar tasks enhances focus and minimizes distractions from constant notifications.
• Perfectionism leads to either paralysis or burnout, both detrimental to academic progress.
• Many PhD-related tasks contribute to busyness rather than actual productivity.
• Academic CVs benefit more from published papers than organizing symposia or events.
• Understanding when to focus on high-energy tasks is crucial for effective time management.
• Collaborating selectively on papers can prevent wasted time and effort.
• Automation tools can assist with repetitive tasks, improving overall efficiency.
• Prioritizing two anchor tasks allows flexibility to manage smaller tasks throughout the day.
• Procrastination can sometimes reveal the true importance of tasks that demand attention.
• Multitasking leads to distraction; focusing on one task at a time enhances productivity.

INSIGHTS:

• Maladaptive perfectionism can create rules that stifle productivity and complicate time management.
• Prioritizing tasks based on energy levels optimizes performance and mitigates burnout risks.
• Batching tasks helps maintain focus and prevents the distraction of constant interruptions.
• Saying no strategically can protect time for significant academic responsibilities.
• Collaboration should be selective to ensure contributions lead to tangible outcomes.
• Understanding the nature of perceived urgency can help manage time effectively.
• Focusing on meaningful tasks rather than busywork is essential for PhD success.
• Recognizing the impact of perfectionism on decision-making can alleviate procrastination.
• Setting boundaries with supervisors is necessary for maintaining academic priorities.
• Building self-awareness around personal energy levels can enhance task management.

QUOTES:

• “Maladaptive perfectionism causes you to create loads of rules for yourselves.”
• “Perfectionism tends to lead to either complete paralysis or burnout.”
• “We love saying ohoh I’m so busy because Society has told us if you’re busy you’re important.”
• “You need to work out when you have the most energy and work with your own daily energy flux.”
• “Saying no more often means that you’re going to free up time to focus on what you should be doing.”
• “A lot of times you can say no I don’t have time for that that is not my priority.”
• “Sometimes the loud things in your mind aren’t the important things.”
• “Batching means that you can focus on one task and if someone really wants to get hold of you, they’ll be able to find you.”
• “Turn off your notifications, don’t look at emails and look at them maybe mid-morning and after lunch.”
• “Multitasking is a massive pain in the bum bum.”
• “You’ll find your time management skills and your productivity will go through the roof.”
• “If you only have to do three of them or even two of them, which two would you choose?”
• “Understanding who is actually true to their word is essential in academic collaborations.”
• “Perfectionism creates narratives about what will happen if we don’t follow these rules.”
• “It’s very rare that an urgent email comes in where you need to respond to this in the next 5 seconds.”

HABITS:

• Prioritize one significant task in the morning and another in the afternoon daily.
• Use energy levels to determine the timing of focused, demanding tasks.
• Batch similar tasks together to minimize distractions and improve efficiency.
• Respond to emails at designated times rather than immediately to enhance focus.
• Say no to non-prioritized requests to protect valuable time for essential tasks.
• Turn off notifications to reduce interruptions during important work sessions.
• Write down and prioritize tasks to manage workload and reduce overwhelm.
• Automate repetitive tasks using available AI tools to enhance productivity.
• Test the limits of personal rules and challenge perfectionist tendencies regularly.
• Create a structured schedule that accommodates varying energy levels throughout the day.
• Collaborate selectively on projects to ensure productive outcomes without wasted effort.
• Establish clear boundaries with supervisors regarding task management and priorities.
• Focus on completing significant tasks before engaging in smaller, low-energy activities.
• Acknowledge and challenge perfectionist thoughts that lead to procrastination or stress.
• Set specific times for reading and writing to ensure dedicated focus on these tasks.
• Reflect on daily accomplishments to build self-esteem without falling into perfectionism traps.

FACTS:

• Many PhD tasks contribute to busywork rather than real academic progress.
• Perfectionism can lead to significant burnout or paralysis in decision-making.
• Academic success is more significantly impacted by published papers than organizing events.
• Time management issues often stem from societal pressures to appear busy and important.
• Batching tasks can improve overall productivity and reduce mental fatigue.
• Energy levels throughout the day influence the effectiveness of task completion.
• Responding to emails immediately often leads to unnecessary distractions and lost focus.
• Collaboration is vital, but not all requests lead to fruitful academic outcomes.
• Saying no is necessary to prioritize tasks that genuinely advance academic careers.
• Perfectionism often creates unrealistic expectations that can hinder performance and well-being.
• Automation tools can significantly enhance efficiency in managing academic workloads.
• Daily productivity can improve by focusing on only one or two significant tasks.
• Multitasking is generally counterproductive and should be avoided for better focus.
• Maintaining a structured schedule can alleviate the chaos of academic life.
• Understanding personal working styles can help optimize time management strategies.
• Prioritizing meaningful tasks helps cultivate a more fulfilling academic experience.

REFERENCES:

• The Five Habits for Monster PhD productivity video mentioned by the speaker.
• AI tools for academics discussed in the context of automation and efficiency.
• Various techniques for managing time and tasks effectively in academia.

ONE-SENTENCE TAKEAWAY

Maladaptive perfectionism significantly impairs time management for PhD students, but strategic prioritization can enhance productivity.

RECOMMENDATIONS:

• Challenge perfectionist tendencies by sending emails without excessive revisions or delays.
• Schedule high-energy tasks for when personal energy levels are at their peak.
• Prioritize significant daily tasks and let less important ones fade into the background.
• Use batching for emails and tasks to maintain focus and prevent distraction.
• Practice saying no to non-essential requests to protect academic priorities and time.
• Automate repetitive tasks using AI tools to streamline workflow and enhance efficiency.
• Reflect on daily achievements to build confidence without succumbing to perfectionism.
• Create a structured routine that accommodates personal energy fluctuations throughout the day.
• Collaborate selectively to ensure efforts yield meaningful academic contributions.
• Turn off notifications during focused work sessions to minimize interruptions and distractions.
• Be selective about which additional responsibilities to accept based on career goals.
• Test personal limits regarding task completion to overcome perfectionist fears.
• Set aside dedicated time for reading and writing to ensure consistent progress.
• Recognize the difference between urgent and important tasks to manage priorities effectively.
• Embrace the idea that perceived failures often have minimal impact on overall performance.
• Foster an understanding that busywork does not equate to productivity in academic settings.

SUMMARY

Professor Terence Tao discusses the transformative impact of AI on mathematics, sharing historical context and modern applications at the IMO.

IDEAS:

• Terence Tao began participating in the IMO at just 11 years, showcasing extraordinary talent early.
• AI tools like AlphaGeometry are revolutionizing how mathematics is approached and solved today.
• Machine assistance in mathematics has historical roots, dating back thousands of years to the abacus.
• Computers have been used in mathematics for about 300-400 years, evolving from mechanical to electronic.
• The term “computer” originally referred to human calculators, particularly during World War II.
• The Online Encyclopedia of Integer Sequences is a valuable resource for identifying mathematical patterns.
• Scientific computation has been used since the 1920s, with early work done by Hendrik Lorentz.
• AI tools now assist in complex mathematical problems that were previously too tedious for humans.
• SAT solvers can analyze logic puzzles and complex statements, but they struggle with scalability.
• AI assistance has enabled the proof of long-standing mathematical conjectures, like the Pythagorean triple problem.
• Formal proof assistants are improving the verification of mathematical arguments and proofs.
• The Four Color Theorem was one of the first major proofs aided by computer assistance.
• Machine learning has recently been applied to discover connections in knot theory and other areas.
• Large language models like GPT-4 can provide solutions to specific mathematical problems, albeit with limitations.
• Formalizing proofs in AI environments can speed up the process of verification and collaboration among mathematicians.
• Collaborative projects using AI have enabled faster and more efficient formalization of complex mathematical proofs.
• The future of mathematics may involve using AI to solve multiple problems simultaneously rather than one at a time.
• Machines can assist in generating conjectures based on large datasets, potentially leading to new discoveries.
• AI’s role in mathematics will remain supportive, enhancing human creativity rather than replacing it.
• Personal interactions and serendipity often lead to new research ideas among mathematicians.
• The integration of AI into mathematics requires mathematicians to retain foundational knowledge to guide AI effectively.

INSIGHTS:

• AI’s integration into mathematics may redefine the boundaries of research and problem-solving methods.
• Historical context reveals that the intersection of machines and mathematics is not a new phenomenon.
• Collaborative mathematical projects can thrive when AI tools assist in the formalization and verification processes.
• Future mathematics could involve large-scale problem exploration facilitated by AI’s computational power.
• Machine learning’s ability to highlight connections in data can lead to innovative mathematical conjectures.
• The evolution of proof assistants has made formal verification more accessible to mathematicians today.
• Humans still play a crucial role in interpreting AI-generated insights and conjectures in mathematics.
• Mathematics is increasingly becoming a collaborative and interdisciplinary field due to technological advancements.
• Serendipity and conversation remain pivotal in shaping research directions in mathematics.
• The potential for AI to automate conjecture generation represents a significant frontier for mathematical exploration.

QUOTES:

• “I hope we all had fun, not just in the competition whether you get a good score or not.”
• “Instead of having three hours to solve a problem, you take months and sometimes you don’t solve it.”
• “We’ve actually been using computers and machines to do mathematics for a long time.”
• “The basic unit of computational power at the time was not the CPU, it was the kilgirl.”
• “In mathematical research, we rely on tables – we call them databases now.”
• “Many promising productive research projects have come up that way.”
• “We’ve been doing scientific computation since the 1920s.”
• “The proof required a few years of computation and it generated a proof certificate.”
• “The future is going to be really exciting.”
• “This may be my most important result to date – better be sure it’s correct.”
• “Every little bubble corresponds to some statement and you don’t need to understand the whole proof.”
• “We’re beginning to sort of prove things that are like 4 or 5 lines long.”
• “AI assistance has enabled the proof of long-standing mathematical conjectures.”
• “I think the future will require more flexibility in research topics.”
• “The hope is that AI will become very good at generating good conjectures.”
• “We still use tables today; we call them databases now, but they’re still the same thing.”

HABITS:

• He reflects fondly on his experiences at the IMO, emphasizing the importance of enjoyment in competition.
• Tao suggests that successful mathematicians often rely on strong mentorship throughout their education.
• Engaging in conversations at conferences can spark new research ideas and collaborations.
• He believes in taking research topics one at a time rather than rushing into multiple areas.
• Tao emphasizes the importance of being flexible in research topics and adapting to new ideas.
• He collaborates with diverse teams, including non-mathematicians, to tackle complex problems.
• Utilizing modern proof assistants has become a regular practice for verifying complex mathematical arguments.
• Tao experiments with AI tools to explore new techniques and approaches in his research.
• He encourages others to learn from mistakes and adapt their strategies when faced with challenges.
• Maintaining foundational knowledge in mathematics is crucial for effectively guiding AI tools.

FACTS:

• The first participant in the IMO to receive a gold medal was Terence Tao at age 13.
• The abacus is one of the earliest machines used for mathematical calculations, dating back thousands of years.
• Computers for mathematical computation have existed in various forms for about 300-400 years.
• The Online Encyclopedia of Integer Sequences contains hundreds of thousands of integer sequences.
• The first major computer-assisted proof was the Four Color Theorem, proven in 1976.
• Scientific computation has been utilized since the 1920s, often involving large human computing teams.
• The proof of the Pythagorean triple problem required a massive computation and was computer-assisted.
• Formal proof assistants are increasingly being used to verify complex mathematical arguments.
• Large language models can provide mathematical solutions, but their accuracy is often limited.
• Machine learning has recently been applied to discover connections between different areas of mathematics.
• The integration of AI in mathematics is projected to enhance collaboration and problem-solving efficiency.
• Collaborative projects in mathematics are becoming more common, often involving interdisciplinary teams.
• The proof of the Kepler conjecture was formalized and completed in 2014 after many years of work.
• Recent advancements in proof assistants have made formal verification processes more efficient and user-friendly.
• AI tools can assist mathematicians by generating conjectures based on large datasets and patterns.
• Mathematics is becoming more collaborative, with mathematicians increasingly sharing ideas and insights.

REFERENCES:

• AlphaGeometry, a tool by DeepMind for answering geometry questions in competitions.
• Online Encyclopedia of Integer Sequences (OEIS), a database of integer sequences.
• Formal proof assistants like Lean and Coq for verifying mathematical arguments.
• The Flyspeck project, which formalized the proof of the Kepler conjecture.
• GitHub Copilot, an AI tool that suggests lines of code for formal proofs.
• The Four Color Theorem, one of the earliest computer-assisted proofs.
• The Birch and Swinnerton-Dyer conjecture, discovered through extensive data tables.
• Condensed mathematics, a field developed by Peter Scholze focusing on functional analysis.
• Various software tools that facilitate collaborative proof formalization projects.
• Notable mathematical events and conferences where ideas and research are shared.

ONE-SENTENCE TAKEAWAY

AI is transforming mathematics by enhancing problem-solving capabilities and facilitating collaborative research among mathematicians.

RECOMMENDATIONS:

• Embrace AI tools to enhance mathematical problem-solving and explore new research avenues.
• Actively participate in mathematical conferences to foster collaboration and share ideas with peers.
• Leverage formal proof assistants to streamline the verification process of complex mathematical proofs.
• Engage with interdisciplinary teams to solve complex mathematical problems effectively.
• Experiment with machine learning to discover unexpected connections in mathematical data.
• Approach mathematical research with flexibility, being open to changing topics and ideas.
• Utilize collaborative project management techniques to break down large proofs into manageable tasks.
• Maintain foundational knowledge in mathematics to effectively guide AI and machine learning tools.
• Seek mentorship throughout educational and research journeys to gain valuable insights and guidance.
• Keep a record of successful problem-solving techniques to reference in future research endeavors.

SUMMARY

The speaker discusses custom tools for LLMs, emphasizing their importance in agent building and functionality.

IDEAS:

• Custom tools are essential for maximizing the effectiveness of LLMs in various applications.
• Tools can be categorized into information retrieval, verification, action-taking, and manipulation types.
• Relevant information gathering can utilize RAG, searches, and databases to enhance LLM performance.
• Verification tools check the inputs and outputs of LLMs, ensuring data integrity and accuracy.
• Action tools empower agents to perform tasks like filling forms or sending messages autonomously.
• Custom tools have evolved beyond simple API calls to more sophisticated interactions with LLMs.
• Clear naming and descriptions of tools are crucial for effective communication with LLMs.
• LLM outputs need structured handling to prevent confusion and inefficiencies in data processing.
• Tools should be designed to handle unexpected or erroneous inputs from LLMs gracefully.
• Building a library of custom tools aids in project efficiency and consistency over time.
• Tools for data retrieval include scrapers, API wrappers, and search engines for information gathering.
• Data manipulators transform LLM outputs into usable formats for further processing or actions.
• Action-taking tools can automate interactions with external systems, enhancing agent functionality.
• Verification checkers can validate code and outputs generated by LLMs for correctness.
• Addressing stochastic behavior in LLMs is essential for managing unpredictable outputs.
• Developing defaults in tool functions helps manage missing or extra input parameters effectively.

INSIGHTS:

• Custom tools enhance agent capabilities by allowing seamless interaction between LLMs and external systems.
• Properly structured tool functions can mitigate issues arising from LLM-generated input errors.
• A well-documented library of tools streamlines project workflows and enables better collaboration.
• The clarity in naming and describing tools directly impacts LLM’s decision-making efficiency.
• Action tools are pivotal in bridging the gap between LLM capabilities and real-world applications.
• Emphasizing verification processes strengthens the reliability of LLM outputs in various contexts.
• The design of custom tools should prioritize user-friendliness and intuitive interaction patterns.
• Tools must adapt to handle the stochastic nature of LLM outputs, ensuring robust performance.
• Effective communication between tools and LLMs can lead to more successful agent interactions.
• Continuous improvement of tool libraries fosters innovation and adaptability in agent development.

QUOTES:

• “This is the secret source of agents.”
• “Custom tools have gone far beyond this concept.”
• “You want your tool to sit in the middle.”
• “You want to make things that are going to be useful for you.”
• “LLMs are stochastic.”
• “You need to tell the agent about the tool.”
• “You really want to make things clear in the name alone.”
• “You want to set up your code to be able to handle these kinds of issues.”
• “Build up your own library of custom tools.”
• “Tools are essential to building anything with agents.”
• “The clarity in naming tools directly impacts LLM’s efficiency.”
• “Tools should handle unexpected inputs gracefully.”
• “Action tools empower agents to perform tasks autonomously.”
• “Clear naming and descriptions are crucial for effective communication.”
• “Verification tools ensure data integrity and accuracy.”
• “You want to ensure that you’ve got tools that work well with LLMs.”
• “The design of custom tools should prioritize user-friendliness.”
• “Managing unpredictable outputs is essential for tool effectiveness.”

HABITS:

• Document tools clearly for better understanding and future reference.
• Build and maintain a library of custom tools for consistent project use.
• Regularly review and update tool functionalities to adapt to new needs.
• Utilize structured naming conventions for ease of use and clarity.
• Prepare tools to handle unexpected inputs to minimize errors.
• Establish default values in tools to ensure functionality despite missing data.
• Engage in continuous learning about new tools and frameworks available.
• Create succinct descriptions for tools to aid in effective communication.
• Test tools frequently to ensure they perform as expected.
• Collaborate with team members to share insights and improve tool design.

FACTS:

• Custom tools are vital for effective LLM application in various contexts.
• LLM outputs can often include unexpected errors due to their stochastic nature.
• Structuring tools effectively can prevent confusion in LLM data processing.
• A library of custom tools enhances project efficiency and consistency.
• Verification tools are widely used for checking the accuracy of LLM outputs.
• Action tools can automate interactions with external systems and databases.
• Clarity in tool naming significantly impacts LLM decision-making processes.
• Proper documentation of tools aids in collaborative development and usage.
• Handling unexpected inputs is crucial for maintaining tool functionality.
• The evolution of custom tools has increased their complexity and capability.

REFERENCES:

• AutoGen
• crewAI
• PhiData
• LangGraph
• PAL model
• ReACT
• LangChain

ONE-SENTENCE TAKEAWAY

Custom tools are essential for enhancing the functionality and effectiveness of LLMs in various applications.

RECOMMENDATIONS:

• Develop clear and structured naming conventions for custom tools to enhance usability.
• Regularly update and refine custom tools based on project needs and feedback.
• Create comprehensive documentation for tools to facilitate understanding and collaboration.
• Implement verification processes to ensure the accuracy of LLM-generated outputs.
• Build a versatile library of tools to streamline workflows across multiple projects.
• Design tools to handle stochastic errors from LLMs to maintain robustness.
• Engage in continuous learning to stay informed about new tool developments and frameworks.
• Prioritize user-friendly interfaces in tool design to improve interaction with LLMs.
• Test tools rigorously to ensure they function correctly in various scenarios.
• Encourage team collaboration to share insights and improve tool effectiveness.

Just came across this probability puzzle and try it on GPT, it uses MLE to solve it and gives a very poor estimate. The question goes like this

A bus company with bus route number starting from . Say we came across one bus with the number 60. What is a good estimate of the number of bus routes .

If we use maximum likelihood estimation, we can consider maximizing

assuming that the distribution is uniform

Since the observation is 60, has to be at least 60. Maximizing gives us 60.

This of course is a very bad estimate, intuitively, the chance of coming across the bus with the largest bus number is rather unlikely. And considering the uniform distribution, it is reasonable to assume that we come across the average instead. Consequently, since there are 59 numbers below 60, we expect .

A similar trick can be used in a more complex case. Assume that instead of observing one bus, we observe 5 buses and the largest number among all is 60. What will be a good estimate of now?

Since , we can consider the five numbers are uniformly spread to 12, 24, 36, 48, and 60. And thus should be . This is a quick estimate but is actually quite good. When , the average of the largest among 5 buses is actually 59.66, which can be estimated by the Lea code below

import lea

bus=lea.vals(*range(1,72))

bus1=bus.new()
bus2=bus.new()
bus3=bus.new()
bus4=bus.new()
bus5=bus.new()

lea.max_of(bus1,bus2,bus3,bus4,bus5).mean()

For arbitrary number of buses, we can compute the average with the function below

def average_max(N,d):
    T=0;
    for n in range(1,N+1):
        T+=n*(n**d-(n-1)**d)
    return T/(N**d)

where the code is quite easy to verify noting that the prob of getting max equal to for buses is .

Self-supervised learning from Meta claimed to beat state of the art without fine tuning.

Continue from part 1, I input

“revise: I spent some fun (and frustration) with gpt-4 for an hour. For your information, the correct answer should be 1/9 and 11/18. When randomly breaking into pieces. There is a nice formula that the expected length of the k-th shortest piece is . gpt-4 cannot get the correct length even for the shortest piece. But the approach is correct. At the beginning, it seems to keep insisting its incorrect answer 1/4. And eventually it gave up, but it throws solutions that apparently violates it original assumption (x<y). And at one point, I commented that its specified region is incorrect, but it just repeated its incorrect region. Maybe I was not specific enough and it misunderstood.”

thinking to let gpt-4 to revise my English for my previous post. Instead, it output

Thank you for your feedback and for sharing the correct answer and formula. I apologize for the confusion and frustration that my earlier responses may have caused. Your insights will help improve the model’s understanding and performance in similar problems.

It is important to note that the ChatGPT model, like any AI model, has its limitations and might not always provide the correct answers. We appreciate your patience and understanding as we strive to improve the model.

The correct answer, as you mentioned, is 1/9 and 11/18 for the average lengths of the shortest and longest pieces, respectively. The formula you provided for the expected length of the k-th shortest piece when randomly breaking into n pieces is a valuable insight:

.

User: given the correct minimum length of 1/9, can you try to rewrite the solution again?

Given that the correct average length of the shortest piece is 1/9, let’s revisit the solution for the problem.

When a 1-meter stick is broken randomly at two places, we have three segments with lengths x, y-x, and 1-y, where 0 < x < y < 1.

We need to find the expected value E[Z_min], where Z_min is the length of the shortest piece. To do this, we will calculate the integral of the probability density function for each possible ordering of the segments and then sum them up.

Case 1) x < y-x < 1-y In this case, we have 0 < x < y < 1/2. The integral for this case is: ∬ x * f_{X,Y}(x,y) dx dy, with integration limits x from 0 to y and y from 2x to 1/2.

Case 2) x < 1-y < y-x In this case, we have 1/3 < x < y < 2/3. The integral for this case is: ∬ x * f_{X,Y}(x,y) dx dy, with integration limits x from 1/3 to y and y from 1/3 to 2/3.

The sum of these integrals will give us the expected value of the minimum length:

E[Z_min] = Integral_Case_1 + Integral_Case_2

After solving these integrals, we find that E[Z_min] = 1/9, which is approximately 0.1111 meters. This is the average length of the shortest piece when a 1-meter stick is broken randomly at two places.

User: A two meter stick is broken at random in two places. What are the average lengths the shortest and the longest pieces?

When a 2-meter stick is broken randomly at two places, we have three segments with lengths x, y-x, and 2-y, where 0 < x < y < 2.

We need to find the expected value E[Z_min] for the shortest piece and E[Z_max] for the longest piece, where Z_min and Z_max are the lengths of the shortest and longest pieces, respectively. To do this, we will calculate the integral of the probability density function for each possible ordering of the segments and then sum them up.

Case 1) x < y-x < 2-y In this case, we have 0 < x < y < 1. The integral for this case is: ∬ x * f_{X,Y}(x,y) dx dy, with integration limits x from 0 to y and y from 2x to 1.

Case 2) x < 2-y < y-x In this case, we have 2/3 < x < y < 4/3. The integral for this case is: ∬ x * f_{X,Y}(x,y) dx dy, with integration limits x from 2/3 to y and y from 2/3 to 4/3.

The sum of these integrals will give us the expected value of the minimum length:

E[Z_min] = Integral_Case_1 + Integral_Case_2

After solving these integrals, we find that E[Z_min] = 2/9, which is approximately 0.2222 meters. This is the average length of the shortest piece when a 2-meter stick is broken randomly at two places.

Now, to find the average length of the longest piece, we can use the following observation: the sum of the average lengths of the shortest, middle, and longest pieces must equal the total length of the stick (2 meters). Since we already found the average length of the shortest piece and we know that the average length of the middle piece is the same due to symmetry, we can calculate the average length of the longest piece as follows:

E[Z_max] = Total Length – 2 * E[Z_min] = 2 – 2 * (2/9) = 10/9, which is approximately 1.1111 meters.

So, the average length of the longest piece when a 2-meter stick is broken randomly at two places is approximately 1.1111 meters.