Annoying calculations: Beta integral

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Annoying calculations: Binomial Distribution

Here, we derive the expectation, variance, and moment generating function for the binomial distribution.

Video, when available, will be posted below.

Why the binomial coefficients are integers

The video, when ready, will be posted below.

Normal distribution: annoying calculations

I am thinking about doing a series of videos on the annoying but necessary calculations one encounters in basic calculus based statistics classes. I made the first video, which I am posting here. Below I post the whiteboards (too lazy to typeset).

The video will be posted as soon as it is ready.

Teaching double integrals: why you should *always* sketch the region

The problem (from Larson’s Calculus, an Applied Approach, 10’th edition, Section 7.8, no. 18 in my paperback edition, no. 17 in the e-book edition) does not seem that unusual at a very quick glance:

if you have a hard time reading the image. AND, *if* you just blindly do the formal calculations:

which is what the text has as “the answer.”

But come on. We took a function that was negative in the first quadrant, integrated it entirely in the first quadrant (in standard order) and ended up with a positive number??? I don’t think so!

Indeed, if we perform which is far more believable.

So, we KNOW something is wrong. Now let’s attempt to sketch the region first:

Oops! Note: if we just used the quarter circle boundary we obtain

The 3-dimensional situation: we are limited by the graph of the function, the cylinder and the planes ; the plane is outside of this cylinder. (from here: the red is the graph of

Now think about what the “formal calculation” really calculated and wonder if it was just a coincidence that we got the absolute value of the integral taken over the rectangle

Commentary

It is not a surprise that my posts here have fallen off by quite a bit. Part of it is that I’ve been assigned a lot of teaching of courses that I have to do extra preparation for.

And yes, the nature of the job has changed. Student body is more or less the same size, but we’ve been reduced from 15 tenure track lines to 9. This means: larger size sections, more students to advise (per professor) and less selection among upper division classes.

Ok, jobs change and I am not treated that poorly. Yes, I wish our classrooms had better IT structure and we didn’t have so many per section.

But the real issue is the loss of tenure track lines and the reliance on visitors.

The number of applicants, at least for the temporary jobs, has dropped by an order of 10.

What is going on, I think: university administrations have embraced, IN PART, a “run it like a business” attitude.

So, they see small class sizes as wasted resources. They see work week “down time” to think about scholarship OR to think about how to approach/teach a class as wasted time. I can’t remember the time I last thought about mathematics during an academic year.

And frankly, I am often not tempted: IF one gets a bit of time to do math, subsequent open slots are often filled by duties, student needs (more students = more exceptional cases; e. g. accidents, illnesses, etc.) and by the time I can get back to it, I’ve often forgotten my train of thought.

And so, what is offered to those who apply for our jobs?
Mostly, it is packed schedules, packed classrooms and not much job security (yes, tenure track faculty can be let go for reasons other than performance).

And back to the “run it like a business model”: we want business demands on potential workers, but we don’t want to compete with respect to compensation.

In my day: I took less money than I could have made outside, in return for being able to do a bit of scholarship and have a bit of fun with classes and enjoy a “family like” atmosphere with extra job security.

That has been taken away; only the “lower pay” remains, and new job seekers are savvy enough to see that.

If we want to “run it like a business”, we need to compete with businesses for young talent, and we aren’t doing that.

My guess: things will be very grim over the next half-decade to a decade, at least at the non-R1 universities.

A “weird” implicit graph

I was preparing some implicit differentiation exercises and decided to give this one:

If find That is fairly straight forward, no? But there is a bit more here than meets the eye, as I quickly found out. I graphed this on Desmos and:

What in the world? Then I pondered for a minute or two and then it hit me:

which leads to families of lines with either slope 1 or slope negative 1 and y intercepts multiples of

Now, just blindly doing the problem we get which leads to: by both the original equation and the circle identity.

Integrals of functions with nice inverses

This idea started as a bit of a joke:

oh ok! integrate this for me please!explain each step of your working too as if you were teaching someone who didn't know calculus at all. should have no problem doing that if it's not hard. thanks!! xox pic.twitter.com/VLdVtjIxRS

— Light (@lightediand) September 12, 2021

Of course, for readers of this blog: easy-peasy. so the integral is transformed into and so we’ve entered the realm of rational functions. Ok, ok, there is some work to do.

But for now, notice what is really doing on: we have a function under the radical that has an inverse function (IF we are careful about domains) and said inverse function has a derivative which is a rational function

More shortly: let be such that then:

gets transformed: and then and the integral becomes which is a rational function integral.

Yes, yes, we need to mind domains.

Vaccine efficacy wrt Hospitalization

I made a short video; no, I did NOT have “risk factor”/”age group” breakdown, but the overall point is that vaccines, while outstanding, are NOT a suit of perfect armor.

Upshot: I used this local data:

The vaccination rate of the area is slightly under 50 percent; about 80 percent for the 65 and up group. But this data doesn’t break it down among age groups so..again, this is “back of the envelope”:

or about 77 percent efficacy with respect to hospitalization, and or 93.75 percent with respect to ending up in the ICU.

Again, the efficacy is probably better than that because of the lack of risk factor correction.

Note: the p-value for the statistical test of vaccines have no effect on hospitalization” vs. “effect” is

The video:

So, you want our tenure-track academic math job…

Background: we are a “primarily undergraduate” non-R1 institution. We do not offer math master’s degrees but the engineering college does.

Me: old full professor who has either served on or chaired several search committees.

I’ll break this post down into the two types of jobs we are likely to offer:
Tenure Track lecturer

Tenure Track Assistant Professor.

Lecturer

No research requirement; this job consists of teaching 12 hour loads of lower division mathematics classes, mostly “business calculus and below”; college algebra and precalculus will be your staples. There will be some service work too.

What we are looking for:

Evidence that you have taught lower division courses (college algebra, precalculus, maybe “baby stats”) successfully. Yes, it is great that you were the only postdoc asked to teach a course on differentiable manifolds or commutative ring theory but that is not relevant to this job.

So hopefully you have had taught these courses in the past (several times) and your teaching references talk about how well you did in said courses; e. g. students did well in said courses, went on to the next course prepared, course was as well received as such a course can be, etc. If you won a teaching award of some kind (or nominated for one), that is good to note. And, in this day and age..how did the online stuff go?

Teaching statement: ok, I am speaking for myself, but what I look for is: did you evaluate your own teaching? What did you try? What problems did you notice? Where could you have done better, or what could you try next time? Did you discuss your teaching with someone else? All of those things stand out to me. And yes, that means recognizing that what you tried didn’t work this time…and that you have a plan to revise it..or DID revise it. This applies to the online stuff too.

Diversity Statement Yes, that is a relatively new requirement for us. What I look for: how do you adjust to having some cultural variation in your classroom? Here are examples of what I am talking about:

We usually get students from the suburbs who are used to a “car culture.” So, I often use the car speedometer as something that gives you the derivative of the car’s position. But I ended up with a student from an urban culture and she explained to me that she and her friends took public transportation everywhere…I had to explain what a speedometer was. It was NOT walking around knowledge.

Or: there was a time when I uploaded *.doc files to our learning management system. It turns out that not all students have Microsoft word; taking a few seconds to make them *.pdf files made it a LOT easier for them.

Other things: not everyone gets every sports analogy, gambling analogy (cards, dice, etc.) so be patient when explaining the background for such examples.

Also: a discussion on how one adjusts for the “gaps” in preparation that students have is a plus; a student can place into a course but have missing topics here and there. And the rigor of the high school courses may well vary from student to student; some might expect to be given a “make up” exam if they do poorly on an exam; another might have been used to be given credit for totally incorrect work (I’ve seen both).

Also: if you’ve tutored or volunteered to help a diverse group of students, be sure to mention that (e. g. maternity homes, sports teams, urban league, or just the tutoring center, etc.)

Transcript: yes, we require it, but what we are looking for is breath for the lecturer’s job: the typical is to have three of the following covered: “algebra, analysis, topology, probability, statistics, applied math”

Cover letter: Something that shows that you know the type of job we are offering is very helpful; if you state that you “want to direct undergraduate research”, well, our lecturer job will be a huge letdown.

Assistant Professor

This job will involve 9-12 hours teaching; 10-11 is typical and we do have a modest research requirement. 2-3 papers in solid journals will be sufficient for tenure; you might not want to have your heart set on an Annals of Math publication. If you do get one, you won’t be with us for long anyway. There is also advising and service work.

What we are looking for: teaching: we want some evidence that you can teach the courses typically taught by our department. This means some experience in calculus/business calculus for our math track, and statistics for our statistics track. For this job, some evidence for upper division is a plus, but not required nor even expected; is is an extra “nice to have.”

But it is all but essential that your teaching references talks about your performance in teaching lower division classes (calculus or below); if all you have is “the functional analysis students loved him/her”, that is not helpful. Being observed while teaching a lower division course is all but essential.

Teaching and Diversity statement : same as for the lecturer job. An extra: did you have any involvement with the math club?

Research: the thing we are looking for is: will you “die on the vine” or not? Having a plan: “I intend to move from my dissertation in this direction” is a plus, as is having others to collaborate with (though collaboration isn’t necessary). Also, a statement from your advisor that you can work INDEPENDENTLY ..that is, you can find realistic problems to work on and do NOT need hand holding, is a major plus. You are likely to be somewhat isolated here. And of course, loving mathematics is essential with us. Not all candidates do..if you see your dissertation as a task you had to do to get the credential then our job isn’t for you.

Another plus: having side projects that an undergraduate can work on is a plus. We do have some undergraduate research but that won’t be the bulk of the job.

Transcript: same as the lecturer job.