Ash

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https://www.ashdoesmath.com

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April 19 2025

Why I Chose Math and Statistics at UofT

For me, choosing to study mathematics and statistics wasn’t so clear; for 18 years, I thought about being a doctor. But what led me to this path, and why did I feel drawn closer to these fields? In this blog post, I’ll share my personal journey and the reasons why math and statistics became my academic passion.

The Fascination with Problem-Solving

From a young age, I was always intrigued by the process of solving problems. Whether it was puzzles, games, or real-life challenges, I found great satisfaction in breaking down complex issues into smaller, more manageable parts. This analytical mindset naturally led me towards studying mathematics and statistics, as both disciplines are deeply rooted in problem-solving.

Math is all about patterns, logic, and structure, while statistics provides the tools to make sense of data, predict outcomes, and analyze real-world phenomena. Together, these subjects offer an ideal foundation for anyone interested in understanding the world through numbers.

The Growing Importance of Data in Today’s World

Another key factor in my decision was the increasing role of data in nearly every industry. In today’s digital age, data is everywhere — from healthcare and finance to technology and social media. Understanding how to collect, analyze, and interpret data is a skill that has never been more valuable.

By studying math and statistics, I felt I could position myself at the forefront of this data-driven revolution. I realized that a solid grasp of statistical techniques and mathematical theories would open doors to a wide variety of careers in fields like data science, machine learning, and research.

The Versatility of a UofT Math and Stats Degree

One of the most attractive aspects of studying math and statistics is the versatility it offers. A degree in these subjects doesn’t just prepare you for one specific career path; it equips you with critical thinking and problem-solving skills that are highly transferable to many professions. UofT offers a specialist combining these two fields, and I immediately knew this was what I wanted to do.

Whether you want to work in finance, healthcare, technology, or government, math and statistics are foundational to understanding complex systems. The ability to work with large datasets, analyze trends, and make data-driven decisions is a sought-after skill in virtually every industry.

The Thrill of Discovering New Insights

What really excites me about math and statistics is the constant opportunity to uncover new insights. Math is a vast and ever-evolving field, with new theories and techniques emerging all the time. Statistics, too, is constantly advancing as new data collection methods and computational tools become available. I am lucky enough to have had professors working on incredible research, and hearing them speak so passionately about them inspires me.

Through my studies, I’ve learned how mathematical models can help explain everything from precious metals to predicting credit card usage. The thrill of applying mathematical and statistical theories to real-world situations is truly rewarding, and I look forward to the many challenges and discoveries ahead in my academic journey.

A Passion for Logical Thinking and Precision

Mathematics has always felt like a natural language for me — one that is clear, logical, and precise. Unlike some other disciplines where answers can be subjective, math and statistics provide a concrete framework for finding solutions. This sense of clarity and certainty is something I value deeply.

Statistics allows for precision in decision-making. Whether it’s determining the effectiveness of a new drug or predicting future trends in the economy, the ability to make informed decisions based on data is incredibly empowering. It’s a field that blends creativity with rigorous analysis, and that’s what makes it so compelling.

The Future of Math and Stats: Endless Possibilities

As technology continues to evolve, the demand for professionals with expertise in math and statistics is only set to increase. The rise of artificial intelligence, big data analytics, and predictive modelling means that the skills I’m developing will be even more relevant in the future.

Whether I end up working in academia, business, or tech, the opportunities available to someone with a background in math and statistics are vast. The ability to understand and analyze data is becoming essential in every aspect of modern life, and I’m excited to be part of this evolving field.

Conclusion: The Perfect Fit for My Interests and Career Goals

Choosing math and statistics at university was a decision that felt right for me. It was driven by my passion for problem-solving, my interest in data, and my desire to be part of a field that has the power to shape the future. As I continue my studies, I’m excited to explore the many ways in which math and statistics can be applied to real-world problems and contribute to society.

If you’re considering a similar path, I encourage you to explore the possibilities that math and statistics offer. Whether you’re drawn to the analytical side of things or the thrill of discovering new insights, there’s a world of opportunities waiting for you in these fields.

April 15 2025

UofT Course Review-STA257

Ash Course Reviews 0

Taken: Fall 2024

Prerequisites/Co-requisites: First year calculus (137/157), linear algebra and multivariable calculus (237/257)

Course Average: B-

How I Did: Above Average

Workload: Average

Content Covered

A mathematically rigorous introduction to probability, with applications chosen to introduce concepts of statistical inference. Probability and expectation, discrete and continuous random variables and vectors, distribution and density functions, the law of large numbers. The binomial, geometric, Poisson, and normal distributions. The Central Limit Theorem.

Lectures

Lectures were twice a week and moderately paced. The professor was incredible; he had clear organized notes as he went through each topic and explained in a very concise way. There were also a few online polls during each lecture, but more on that later. His office hours were also fantastic; he was a great guy to talk to not just about the course but beyond, he’s extremely passionate about what he does.

Tutorials

Each week, we had a tutorial right after the lecture where we went through a few questions related to the week’s topic. To be honest these were a bit hit or miss; some were really useful while others were a bit boring and redundant. But the TAs were good at explaining and had lots of office hours closer to exam times.

Evaluation Methods

Poll Participation, Midterm Exams, Final Exam.

Poll Participation

Every lecture, we had a few polls in the form of multiple-choice questions about questions based on certain topics. We had a few minutes to discuss the question with our neighbours before inputting our answer. He would then go through the question and give the answer. We had a little over 100 questions total by the end of the semester.

Midterm Exams

We had 2 midterms throughout the semester. Each were 100 minutes long and tested material up to that point. They were a mix of computational and proof based, but a little skewed towards computations. The first one was not too bad, all the questions were not bad at all. The second midterm though, well, let’s just say that it wasn’t the best (integrating that absolute value pdf will forever haunt me), but I ended doing better than what I expected.

Final Exam

The final was 3 hours long and was a dead average between the first and second midterms in terms of difficulty. All the questions were very fair, and I am forever grateful that the one topic I did not like had a pretty simple question on it which I was able to do.

My thoughts on the course and how to do well

This is my favourite course taken to date; I love the entire theory of probability and studying the odds of things happening, and this course definitely gave me a strong foundation in that. The one topic I mentioned that I didn’t like was the idea of convergence, more specifically ‘Convergence in Probability,’ which has made a few appearances later in my statistical journey, so I had to get used to it, but now it’s not that bad. Don’t be scared by the description of ‘rigourous’; it does have some faster paced sections, but overall, it’s very manageable if you do all the practice problems and go to all the lectures. An absolutely goated course that I would recommend to anyone who needs stats for their degree.

April 15 2025

5 Things I Wish I Knew Before Starting University

Ash University Tips 0

I remember 20-year-old me showing up on the first day of classes, eager to get the university experience he dreamed of. Now, almost halfway through my undergrad, I can confidently say that things did not go according to plan, but that’s OK.

I learned not only a lot of math and stats in 2 years, but a lot about myself and life in general. Whether you’re a prospective student or just about to graduate, you should read on and see if you relate.

1. You don’t have to have it all figured out.

There’s so much pressure to figure out your major, your residence, making friends having a set plan for when you graduate. But I can confidently say that my plans went straight out the window after the first 2 weeks; I had to modify my program because I realized I really hated overly theoretical material, and my social life was drier than my sense of humour. I was really thrown at first, but I soon learned that sometimes the best plan is having no plan.

I’m not endorsing throwing caution to the wind on every occasion, but be open to other avenues, whether it be academic, social or otherwise. I’m almost finished with my second year, and I can definitely say I’m very happy with how things turned out.

2. You will have bad days.

There will be some days where you’ll be off your game. You may not be able to focus on everything in lecture, you may not be able to participate in tutorials like you usually would (because WE all participate happily and voluntarily) and no matter how much you try to study, nothing seems to stick.

Let me tell you, all of this is normal. Whenever I have a bad day, I firstly always acknowledge that I’m having a bad day and try to just accomplish the necessary things. After that, I do absolutely no work for the rest of the day (I know if you’re a UofT student, this could mean a death sentence depending on your program). But I’m telling you, taking that rest is definitely important for moving forward, which leads to my next point.

3. What works for one person may not work for you.

There is no ‘one size fits all’ approach to anything in life, and anyone who says otherwise is lying. We are all unique human beings at the end of the day, and, while there may be some overlap, we all cope and relax in our own way.

I always hear stories from my friends who pulled all-nighters at Robarts drinking lethal amounts of Red Bull, yet I can’t study past 10:00 without spacing out. While some people take a walk or do sports to relax, I like to write blog posts while listening to random Eurovision songs.

My point is, do what makes you happy and you’ll be refreshed to go again.

4. You’ll meet a lot of people, but you’ll also feel lonely sometimes.

University is a huge place. I can confidently say I have met hundreds, maybe thousands of people over the past 2 years here, but I still feel lonely sometimes. Honestly, I don’t have the stereotypical close-knit friend group you see in college movies, the ones you see having a picnic on a sunny day on the campus grounds. I’m not a social butterfly who clicks with everyone right away, and no matter how hard I try, most don’t seem to work out or last.

Unfortunately, some people can be cliquey and downright awful, but the right people, who appreciate you for you will find you eventually. While this may sound cheesy, it was definitely true in my case; I’ve found a few friends that I am grateful for everyday and who knows who I may meet later down the line.

5. Put yourself out there but don’t lose yourself in the process.

In university, no one tells you what to do. It is up to you and you alone to decide who you want to be later down the line. You may even find yourself changing in some ways; the way you speak, the way you act, who you hang out with and more. This is completely fine and is a fundamental part of growing up, but don’t lose yourself in the process. Don’t lose your quirks, your soul, your identity just because a few other people may not understand it. Realize that you can never please everyone.

This was probably the harshest lesson I learned, because I was so desperate to fit in like everyone else. But I soon realized that I didn’t feel like myself and didn’t like the person I was becoming. So, I took a step back and re-evaluated myself and moved forward. I mentioned I’m not a social butterfly, but I try to make small talk where possible to try to get to know people. I’m still a work in progress, but better than what I was before.

So, there you have it. I’m still learning so much every day, and I hope these lessons would help you out.

February 20 2025

UofT Course Review-ACT230

Ash Course Reviews 0

I’ll be reviewing one of my favourite electives, a finance course for non-actuaries. It gives a general introduction to financial concepts which I think everyone should be at least familiar with since financial literacy is one of the most important life skills. Once again, this is my personal experience and is affected by the exact instructor teaching and specific assessment methods.

Taken: Fall 2024

Prerequisite: Any first-year calculus course

Course Average: C +

How I Did: Above Average

Workload: Average

Content Covered

Introduction to financial mathematics, interest measurement, present value calculation, annuity valuation, loan amortization, consumer financing arrangements, bond valuation.

Lectures

Lectures were once a week where we learned a new concept roughly each week and did a lot of practice questions related to the topic. The instructor was really good; very straightforward, clarified questions with ease and went through all the different tips and tricks with certain topics. The best part was we were told what kind of questions would appear on the exams, which I’ll get to in a bit.

Tutorials

Each week, we had an online tutorial where we went through a few questions related to the week’s topic. The TA’s were awesome at explaining and really broke down questions step by step to get to the answer. I liked that these were online so they could be done from anywhere without the hassle of rushing to class.

Evaluation Methods

Quizzes, Midterm Exams, Final Exam.

Quizzes

Every 2-3 weeks, we had an online quiz with 3 questions related to the previous weeks’ material. These quizzes were single attempt with a 30-minute time limit, so speed and accuracy were important. We had 5 in total, with the best 3 being counted towards our final grade. The questions were on the easier side, being straightforward with the computations and all relevant information given. They were also in the multiple-choice format, so you could plug in and verify the answer.

Midterm Exams

We had 2 midterms throughout the semester. Each were 50 minutes long and tested material up to that point. The first one was 6 long answer questions, which was quite difficult to finish in that time frame. The questions themselves weren’t too bad, but the time was too short in my opinion given we had to show all working. Luckily, I was able to finish in time, but with only 3 minutes to spare at the end to check over. The second one was better, with 3 multiple choice (still had to show all working) and 2 longer questions which were a lot more manageable in 50 minutes. It also helped that we were allowed a formula sheet, which was half of 1 side of an A4 page.

Final Exam

The final was 2 hours long and had 10 questions with some parts. The questions were for the most part straightforward but a few challenging and time-consuming ones. This time, we were allowed a full side of an A4 page for formulae, which was quite helpful (all formulae could fit on 1 side). I finished with quite a bit of time to spare and was able to redo all of the computations, so no complaints.

My thoughts on the course

This was one of the best courses I have taken. I gained a better understanding of basic financial principles such as interest rates, loans and bonds which are very applicable to real life. The formula sheet was a lifesaver since memorizing all the formulas would have been pretty difficult. My favourite topic was loan amortization, since I got to see applications to credit cards (my latest project, you should read it!) and mortgages, giving me a better insight into how monthly payments are calculated (useful in the future if you take out a mortgage/loan to make sure they’re not scamming you). The instructor and TAs were kind and very helpful with a lot of support though office hours and patience answering questions. I highly recommend this course to anyone (it is computation heavy, so be prepared!).

February 18 2025

Love At First Swipe: A brief look into credit cards

Ash Data Analysis 0

Today’s project will be about one of my favourite things: credit cards. I think of them as a debit card with an extra step: instead of the money coming from your checking account, it comes from the bank, and you just pay it off using your checking account. Of course, this assumes account holders have enough self-control as to not spiral out of control and accrue massive debts by overspending. But, if you know how to use them responsibly, they can be one of the best things ever. From perks, rewards to securing bigger lines of credit in the future, I think they’re worthwhile if one can be responsible.

Methods

I created a decision tree model which predicts a person’ s credit utilization ratio is high or low based on different factors such as their credit limit and revolving balance of their account. Most credit bureaus state that a person should not use more than 30% of their available credit, so that’s what I used to create the High-Low split of the model. I then analyzed this model by creating a confusion matrix of predications and assessing the model’s accuracy as well as any limitations. I also did a financial evaluation using credit card interest rates and an amortization schedule for paying off credit card debt to see just how long it takes.

Analysis

The graph shows that there is an overall negative relationship between credit limits and average utilization ratios. This is unsurprising a fixed amount spent with a lower limit will have a higher ratio compared to a higher limit.

The decision tree created had an accuracy of 96.8% when classifying new data, which is quite high. The corresponding confusion matrix yielded a false positive rate of 3.58% and a false negative rate of 2.45%, both very low when compared to the sensitivity and specificity, both above 95%. This implies that the model is a very good predictor of whether someone has a high or low credit usage based on their credit limit and total revolving balance at any given point in time. However, it is important to note that this model would need further testing on other datasets to confirm its applicability as it may conform to this data, but not to others.

Based on the amortization, we see that it takes 35 months, or approximately 3 years, to pay off a $5000 debt assuming consistent monthly payments of $200 are made at the end of each month. This is a quite some time which assumes that a person can put aside that money for this specific purpose. We also see that as more of the principal is repaid, the lower the interest accrued becomes, suggesting a negative association.

If the monthly payments were higher or payments were made on a different time basis, such as fortnightly, this would take less time as less interest is accrued, so the balance decreases to zero much faster. But if the monthly payments are less of if you miss a month or two, this will set you back even further as more interest accrues.

Conclusion

Credit cards can be very useful if used responsibly. If not, they can accrue massive debts that will take many months, even years, to pay off.

January 4 2025

UofT Course Review- AST101H1

Ash Course Reviews 0

By writing course reviews about classes I took at UofT, I hope to help any prospective students struggling to plan their schedule. There will obviously be some degree of bias as these are based on my own experience and certain factors such as the professors teaching, exact content covered and assessment methods varying from year to year. However, I hope they give some insight into what to expect.

I’ll start off with AST101H1: The Sun and its Neighbours.

Taken: Fall 2023

Course Average: C+

How I Did: Above Average

Workload: Light to Average

Content Covered in AST101

Official Description

Our place in the Universe. Phenomena we see in the sky. What we know about the Sun, the planets and comets, and the formation of the solar system – and how we know it. What makes planets suitable for life. Finding out about the nearest stars and their planets. This course is intended for students with no science or engineering background.

What we covered

Our galactic neighbourhood going outwards (planets, galaxies, clusters, superclusters, the edge of the observable universe). The Earth (formation of our Moon, tidal and seasonal phenomena and our future), star formation, why planets and galaxies form the way they do, the light spectrum, characteristics of our Solar System (the planets, dwarf planets, our Sun, comets and asteroids) including their formation and physical and chemical properties. Space exploration and techniques, methods for identifying planets, and potential for extraterrestrial life.

Lectures

Lectures were in the Convocation Hall since this class started off with 1500 people (yes, that is a LOT) and about 1000 (still a lot) stayed by the end. I sat in the very front row (I’m a nerd, I know) but it helps me to focus more and engage if need be. AST101 is a very popular course for fulfilling science breadth requirements for social science, business and humanities students.

We had two professors for this course, with one for each half of the total material with one or two appearances for subtopics in between. They were both incredible; they didn’t just read off the slides, they spoke with passion and you could clearly tell that they loved what they did. They would walk around during the Clicker activities (see below), seamlessly clarified any doubts people had and even dropped a hint or two. They also showed real examples of data and photos from NASA about recent discoveries and crisp images of the Solar system, which looked beautiful and ties all of the material together. They even had cool animations of orbits and phenomena to help visualize the concepts, which I really appreciated.

Tutorials

Each week, we had a tutorial where we were split into 4 groups and had to collaborate and present a concept from class with 20 minutes to prepare. After each presentation, we had a Clicker question to answer, making it 4 per tutorial. My group mates were very kind and easy to work with, so I have no complaints there. My TA was also pretty good, he was very knowledgeable and explained and clarified any concepts well.

Evaluation Methods

Weekly Quizzes, ‘Clicker’ activities, Experiential Project, Midterm Exam, Final Exam.

Weekly Quizzes

Every week, we had an online quiz with a few questions on that week’s lecture material. We were allowed up to 5 attempts for the quiz, with our most recent score, not the highest score, recorded. Every new attempt does not have all the same questions; questions seemed to have been sampled from a question bank with every attempt. The questions weren’t too hard though, just fill in the blank, multiple choice and select all that apply.

Clicker Activities

We had to buy a physical remote called an ‘IClicker’ before classes started which was used for multiple choice polls every lecture, with there being 2 – 5 polls per class depending on the topic. It was a good way to keep everyone focused and engaged as we were allowed to discuss with our neighbours to bounce ideas off of each other. The only thing that sucked was that the bookstore didn’t have any for the first few weeks of class, so I went on Amazon and bought one. Luckily, they were aware of the issue and did not take any polls for credit until most of the class had, about 3 weeks in. However, I think using software like Poll Everywhere would have been significantly better since people could just download it on any device and respond from there.

Experiential Project

We had to choose between 3 projects to work on for the semester:

1. A project where we everyone took pictures of the Moon over the course of 2 months, and write a research paper on the observations made from those photographs. However, we were allowed to only use concepts from lectures.

2. Take part in two 8-minute oral interviews about any topic of our choosing, where we discuss what we learned and any questions about the topic could be asked.

3. Set up a telescope in public and ask volunteers from the public to observe the sun through it. It had a special filter as to not burn anyone’s retinas and blind them.

I choose the Moon project because I enjoy doing research and analyzing data. I took my Moon photo on the night of a full moon using a camera provided by the faculty. I’m no photography expert by any means, but I managed to get a few good shots that night since it was cloudless, and the Moon was bright. The project had to be fully structured with sections for an introduction, methodology, results and analysis, including a bibliography with all sources used. I’ve attached a copy of my report below so you can take a read of it: Moon Report

Midterm

It was a 2 hour exam with a mixture of multiple choice, short answer questions and one bonus question for extra credit. The exam was very fair with no surprises; it covered all material up to the date.

Final Exam

My final exam experience was a bit unconventional. I did not write the final exam with everyone else; I had something called a ‘3-in-a-row’ conflict where I had 3 exams scheduled back-to-back, so the latest one (this) was deferred until later in the day. I’m grateful because this was an afternoon exam, and I had another one that same morning and another the night before, so I would have been burnt out. Instead I wrote my final that night.

The final was the same format as the midterm with multiple choice, short answers and a bonus question and emphasized material since the midterm but had some concepts from earlier. The only difference was that it was 3 hours long. It was a bit tougher than the midterm but overall, still very fair. However, I’m pretty sure the exam I wrote was not the same as everyone else’s. I never followed up to see if I could get a copy of the final everyone else wrote, but I’d like to think it would have been around the same difficulty.

My thoughts on the course

This was a very interesting class to take for me as I’ve always been interested in Astronomy. Learning about what lies beyond our planet was very fascinating. The professors made all the material digestible, and the concepts were not too hard to learn. A bit of memorization is involved but nothing drastic and the course had little to no computational math involved (as a Math and Stats undergrad it broke my heart). Luckily, this course’s workload was light for me, and I did well while only putting in 1 to 2 hours of effort per week (actual effort may vary).

My favourite topic was the search for potential extraterrestrial life since there is a high chance that we are not alone in the entire universe and seeing that from an objective point of view really intrigued me. All the assessments and workload were very fair and there were plenty of office hours and support available.

Overall, I’d recommend this course to anyone.

January 1 2025

All that Glitters: Analyzing Precious Metals

Ash Data Analysis 0

I’ll be doing a brief analysis of gold and silver investments using sourced data.

Introduction

Gold and silver have been prized by people for thousands of years. One of the oldest forms of currency, they were associated with high value due to their rarity and lustrous appearance. Today, gold and silver have many industrial applications in the production of jewellery, coins and even electronics due to their high electrical conductivity and resistance to corrosion. Also, they are still used as a form of currency but not in the generic sense. Because of their scarcity and demand in industry, they are treated as commodities and can be bought in the physical forms such as coins, bars, bullion or assets such as stocks in mining companies and exchange-traded funds. Investors often buy silver and gold as a hedge against inflation as they appreciate over time. But this raises the question: Are these precious metals worth the hype?

Methodology

Data was sourced from Kaggle, an online platform with free datasets and used this as a basis for my analysis. I then used R coding to clean and manipulate this data to create visualizations and calculate the overall rate of return for both gold and silver separately. I then used this data to form a linear optimization problem with constraints to maximize an initial investment and calculate the amount of silver and gold needed to achieve it. Lastly, I used some financial mathematical techniques to determine how much gold and silver you need to buy per year to accumulate to a fixed amount a decade later.

Graphs

Techniques

You can view the techniques I used by clicking the relevant word below:

R code

Optimization

Payment Valuation

Analysis

Based on the graphs of prices and return rates, we see that both gold and silver had an overall positive increase over the last few decades. There are noticeable spikes for the years 1980 and 2008, with both being attributed to extreme geopolitical and market events (for example, the collapse of the Bretton Woods system which affected the volatility of the U.S. dollar in the 1970s coupled with rampant inflation in the early 1980s and the more recent 2008 Financial Crisis). Nonetheless, the average return rate for silver was 3.96% and 6.57% for gold, which are quite impressive.

Observing the optimization problem, we see that $60000 should be invested in gold and $40000 should be invested in silver to reap the maximum return on investment, found to be approximately $5526 per year, which is not insignificant by any means (You can buy yourself a new wardrobe or take a nice vacation with that!)

Lastly, valuing some payment streams using silver and gold prices showed that you need to buy less gold than silver to get the same value, which makes sense given the higher return rate. But that consistently investing in both silver and gold would lead to hefty returns in the future.

Limitations

1. Quite a few assumptions were used for these models, so actual quantities will vary depending on future expected and unexpected global events.
2. My analysis heavily relied on the rates I found to be accurate, but more data would have painted a better picture.

Conclusion

Based on all the data and analysis above, gold and silver seem to be very good investments, both in the short and long term. Therefore any investor, whether new or veteran, should have these in their portfolio (This is NOT professional advice, so please weigh all risks appropriately if you are serious about investing).