# Proportional density function question

Here is a question I gave on an exam last year. Please let me know what you think of the question (eg: if it is a fair question, easy, difficult, etc). A lot of students were upset for questions like this, but apparently what I am noticing is that they are woefully unprepared for Math coming into the college level, even in so called "higher" Math classes. I can't speak for all colleges, obviously, but that is my impression in several.

The two sides of an isosceles triangle have a length of $$l$$, each side, and the angle between them is the value of a random variable called "$$x$$" with a function of proportional density to $$x(\pi-x)$$ in each point $$x \in \left(0, \dfrac{\pi}{2}\right)$$. Calculate the function of density of the area of the triangle and the mean (expected value) of the random variable.

This is for a Calculus III with Analytic Geometry course (and the requirement is that a student has grade of C or better in the previous course). It is an honors course. Most of the students are majoring in Math education or simply Math. Their expectation is to then go on to Probability I and II, so obviously we cover a range of topics, including Probability and analysis topics.

• It depends on what the students know. Even if you specify the age of the students and the country they are in, what is easy or difficult can differ from one high school to another, even if they are in the same city. Jun 6 '21 at 13:00
• Very true, Joel. This was given in a Math class for "Math education majors", in the US...so you have an idea.
– Wasp
Jun 6 '21 at 13:04
• The wording may also be a bit awkward for U.S. students not especially familiar with the concepts. For example, "in each point" is a bit strange (usually one says "at each point"; note that points have nothing in them) and "of proportional density to" is a bit strange (one usually says "of density proportional to"). Also, would the students have previously been exposed to the idea of a random variable? Calculus-based probability was not required for math majors (even those planning to attend graduate school) where I was an undergraduate, and I didn't take such a probability course until later. Jun 6 '21 at 15:40
• That question would take me some time to figure out. Did you teach problems like it in the course? Jun 6 '21 at 16:47
• The fact that this is an Honors Calculus III course changes things dramatically. Context like that should have definitely been included in the original post. Jun 9 '21 at 0:57

People respond well to perceived effort and investment on the part of others. When I get an email that is worded poorly from a colleague, or start reading a paper that has not been carefully revised, I am more likely to tune out and abandon the situation.

It is possible that you put significant effort into creating this question, but when I read it, it feels like a question that was thrown out to me in first draft form. If a professor gave me this question, I would feel that it is unlikely that the question is even possible, let alone good. Additionally, your opinion of the students is clearly low as shown by your comments, and the students are going to be able to detect that.

I would say the question is basically impossible as written, not because of the math, but because of the social cues above.

Here is what happens when I try to read this question:

The two sides of an isosceles triangle have a length of l, each side,

What is meant by "the two sides of a triangle?" Triangles have three sides. "The" here indicates that these two sides are the only two sides. That's not true. From context I guess you don't care which two sides, but the word "the" starts off the sentence with confusion. Then, what is "each side" doing in this question? It doesn't clarify anything. Maybe you were worried I would think that two of the sides had a total length of $$l$$? Then you would have used the word "total."

I am fourteen words into the question and I have already had to stop three times to figure out what is being asked. As a student, I am somewhat hostile already: the person writing the question did not take much effort to communicate with me. All of this effort could have been avoided if the instructor had drawn a picture of a triangle, or revised the first draft of the question. I do not have high hopes.

the angle between them is the value of a random variable called "x" with a function of proportional density to x(π - x) in each point x ∈ (0, π/2).

What? At this point, in a timed test, I am looking for a different question. This question is either here to try to get me to waste time, or it is here for some other student to get correct -- it's certainly not for me.

Calculate the function of density of the area of the triangle

I do not know what this is and I don't know any student that would know what this is.

If you want to ask students this question, you need to work hard on revising it to communicate what you are asking. Here is a second draft of the first couple of sentences, which at least communicate the triangle you have in mind without requiring significant mental effort on the part of students who already know you don't expect them to get the question correct:

Two sides of an isosceles triangle each have length $$l$$. The angle between these two sides is given by $$x$$, which is between $$0$$ and $$\pi/2$$.

• Agree that the wording is weird. "Function of density of the X" is what I guess is the pdf of X. Word order in "of proportional density" is confusing (implies that there is some special "proportional density"). Using same notation for the variable and its values is confusing. Reference to "the random variable" in the end is ambiguous. But all this probably matters little to the students - the ones who can identify the correct math steps probably can also read between the lines to figure out what is actually being asked.
– juod
Jun 6 '21 at 22:46
• The reason I have bolded the word feel repeatedly is to emphasize the perspective of the reader. It's irrelevant whether you put a lot of time into writing the question, because the feeling the reader gets upon reading it is the important part. It's also irrelevant whether you are popular or loved or reputable. This isn't a personal attack. The question sucks, and it is fixable. Jun 7 '21 at 0:29
• Except that was not what students found hard about the question. I did very much perceive that you were personally attacking me by the tone of your message. I am sorry I will engage with others who do not resort to insulting a question in such a petty way.
– Wasp
Jun 7 '21 at 0:46
• "By the way, I can accept that the question may in fact be poorly worded, but I am almost chuckling because I am known to be extremely well versed in the way I write and how I present my proofs. I accept any criticism, but this is just overkill on your part." No, it's not. The question is incredibly hard to understand. It is hard for Chris Cunningham, and it is hard for me, and presumably it will be even harder for your students than it is for us. Your students might be too polite to say it.
– Stef
Jun 7 '21 at 14:39
• @Wasp "Except that was not what students found hard about the question." Did they tell you what they found hard about it? You say a lot of students were upset for questions like this, but what were they upset about? What was your specific feedback from them? Jun 7 '21 at 16:55

Like most of the respondents here, I had to read this problem several times before I could understand what you are asking.

1. The phrase "a random variable called $$x$$ with a function of proportional density to $$x(\pi - x)$$" is both nonstandard usage and potentially misleading. On my initial read of the problem I thought you were trying to describe the mass density of a triangular lamina, a common problem type in multivariable calculus. But under that interpretation the rest of the problem didn't make any sense (or at least I couldn't make sense out of it).
2. Only after reading the problem a few times did I finally realize that you meant that "$$x$$ is a random variable with probability density function proportional to $$x(\pi- x)$$", and that by "function of density of the area of the triangle" you were asking for the corresponding probability density function of the area.
3. In particular notice that the adjective "proportional" in your sentence is misplaced; it belongs alongside the preposition "to", as in "proportional to", not modifying the word "density" as you have it.
4. Some authors prefer the phrase "probability distribution function" over "probability density function" precisely to avoid the possible linguistic confusion I experienced between the two meanings of the word "density".
5. Who are your students? In your question you refer both to "Math coming into the college level" and "even in so called 'higher' Math classes". But these are not at all the same populations of students. It is not at all reasonable to make inferences about what one group can or should know based on the performance of the other group.
6. Specifically, there is no reason whatsoever to expect students "coming into the college level" to have ever seen probability density functions (pdfs) before. They are not part of the K-12 curriculum and are not commonly taught in lower-level undergraduate mathematics courses. Some textbooks (e.g. Hughes-Hallett) include them in a section on applications of integration in Calc 2, but many universities -- including my own -- routinely skip that application, as there simply isn't enough time in the semester to cover everything in the book.
7. You have not said what your class is on, only that it is a class for Math education majors. Is it a class on probability? Have you done other examples of probability density functions? Or is perhaps a class on probability a prerequisite for students in your class, so that it is reasonable to presume that this is prior knowledge? The wording of your question seems to suggest that you expect students to already know this language, which makes me think that you have not used it before. If this is the case, I would suggest you take a look at your own department's course offerings, especially those typically taken in a sequence prior to the course you teach, and ask yourself where a student would have been likely to encounter this before.

One final thought: various bits and pieces of your phrasing combine to suggest to me that your own undergraduate education was not in the United States, and that English is not your first language. (If I am wrong, I sincerely apologize for the inference.) If that is the case, it might explain a number of related phenomena: your use of nonstandard language, your students' difficulty understanding what you mean, and your surprise at what students do and don't know.

EDITED TO ADD: Incidentally, I was curious about the problem so decided to solve it myself, and found that the probability distribution function of the area is given by the function $$g(A) = \frac{24 \arcsin\left(\frac{2A}{l^2}\right) \left(\pi - \arcsin\left(\frac{2A}{l^2}\right) \right)}{\pi^3 \sqrt{l^4 - 4A^2}}$$ It's possible that some clever identity could be used to simplify this, but if there is, I don't see it. Of course it's also possible that I have made an error in my solution, in which case I hope someone (perhaps the OP?) will post the correct answer. But if I am right, the ugliness of the answer is itself (in my opinion) strong evidence that this is not a reasonable question for an exam.

• Thank you mweiss. I'll start backwards: I do not speak any other language other than English and I have never studied abroad (although I have lived abroad for several years, but that is in no way connected to my Math). All of my formal Math education was here and to a lesser degree in England. Re: Class. It is not a class on Probability, but we have covered this topic. It is a Calculus class for Math ed majors.
– Wasp
Jun 7 '21 at 2:41
• As for the expectation about what they know, this comes from 2 things: the topics I have covered with them in class and 2) my own observations about courses even at the introductory level. I am truly surprised how so many students still are applying almost a "Betty Crocker" type of solution to their Math problems. So, I must have gone to an extremely good high school because there are many topics I cover in class and students honestly tell me they have never done it. I believe them and it's an unfortunate situation.
– Wasp
Jun 7 '21 at 2:42
• Numerical integration shows that my solution, when integrated from $0$ to $l^2/2$ (which is the maximum possible area), yields $1$ for all values of $l$ -- which doesn't prove anything, but is at least consistent, and reassures me that my answer is, at least, a plausible probability distribution function for the area. Jun 7 '21 at 4:51
• I have posted the problem itself, along with my solution, to math.stackexchange.com/questions/4166154/…. Jun 7 '21 at 17:06
• A small nitpick that also applies to the original wording of the question. There is a distinction between a random variable and its value. The probability density of a random variable is not a function of the random variable but of its value. So a more careful phrasing might be "Let $\theta$ be a random variable with a probability density proportional to $x(\pi-x)$, where $x$ is the value of $\theta$." Jun 8 '21 at 6:31

You are asking a lot of them, even though each little bit of the problem isn't especially difficult and was probably covered in a class that they have taken. You need to remember

• a bit of trig/geometry to calculate the area from an angle,
• a bit of statistics/calculus to get the normalization constant and a formula for the expected value, and
• a bit of calculus to do the integration of trig functions.

Each of these topics has been taught to them in a disconnected manner, so simply combining the topics from different courses is a challenge in its own right.

As always, difficulty depends on what you have been teaching them and how much time they had to do it.

• Thank you. This was a timed exam and only one of the questions. You are absolutely correct about the "disconnected manner" this has been taught to them, so I was testing to see if they could make those important connections. Very few did. In fact, in a class of 32 students, not a single one of them got the entire problem correct, unfortunately.
– Wasp
Jun 6 '21 at 17:08
• I was testing to see if they could make those important connections --- Did you sufficiently prepare the students for questions like this? I suspect not, based on what the students said. My experience has been that if you deviate (in a more difficult direction) even slightly from what students expect, then you'll be slammed with complaints. What needs to be done is a lot of prior practice with questions of this sort (even "sample tests" in advance, if you deem the deviations to be sufficiently important) so that the students have no basis on which to complain about "surprise problem types". Jun 6 '21 at 19:08
• Dave: I did. They know the terminology and my style. They complained that it was too hard, not about the terminology itself. They know exactly how I word things, so this is not all that new to them. The only difference is that I mixed several concepts to test if they are keeping up, but we have indeed done all these things.
– Wasp
Jun 6 '21 at 22:14
• User615: I wish they allowed for longer comments here so that I could post at least the main questions on that exam. But, I agree that mapping the topics that way really stumped them in several ways.
– Wasp
Jun 6 '21 at 23:50

To me, this question seems poorly worded.

1. Not sure what "the" two sides of a triangle even means. Do you mean the two equal sides?

2. After that, seems like we get some info about an angle and have to use that to predict info about the area. [Probably using the formula for area. Donno. Guessing.] OK. I'd say, this is medium hardish. Not end of the world. But certainly, not easy drill. I hope you are covering easier drill. Or your students are such brainiacs they don't need that checked.

3. In some ways, this question feels very 19th century to me. Lot of language to translate into math. Then execute the math. Then translate the math back into language. Not saying this is the end of the world, of course. Heck, look how people struggle with basic chem stoichiometry! All that said, I would be leery of pushing this with weaker students or even with strong students who haven't been trained for the game. And I even think there is some benefit to these translations (life is word problems). But...don't trivialize the challenge.

4. It's a mistake that you don't tell us more about the audience (age and "track"). These are key aspects in pedagogy. What's right/helpful for Susan Randall in Harvard is not what's right for a lower track kid in Brooklyn. This is a PART OF THE EQUATION (in pedagogical questions).

5. Cris C. is a sweety. If he's ragging on you, you're hurting. I'm the mean bridge troll.

• Ha! Very accurate, "guest". I have been called "very 19th century" on numerous occasions. My students are a mix. In general, they have me always teach two extremes: either the "honors" Math classes (and this is an honors class) or the remedial, where I am known to be hard but loved because I show them things they never imagined in their lives exist. With regard to the age: I'd say most range in ages from mid 20s to maybe 30s. I agree with all you've said thus far. As for Cris C, my take on his reply to me was that it was unnecessarily rude. Maybe he did not mean it that way, but...
– Wasp
Jun 8 '21 at 1:34
• OK, good luck... Jun 8 '21 at 11:31
• If two sides of a triangle have length $l$, each side, then the triangle is (by definition) isosceles. I don’t see any reasonable pretext to quibble over the point 1. Jun 9 '21 at 9:49

I sketched out a partial solution to this problem on a piece of scratch paper. I could be missing a simplification or getting something wrong, but the following is what I came up with. It took me about a page of math to get the pdf of A, expressed in terms of x. To get the pdf of A, one would then have to invert the function A(x) and substitute, which would be doable but looks messy. After that, one would need to do an integral in order to get the expectation value. I would be pleasantly surprised if that integral came out to be something that could be done in closed form without a huge mess.

(1) This problem seems much too long and messy to use as a single problem on a timed exam.

(2) It requires putting together a lot of knowledge: probability theory, integration, differentiation, and trig manipulation. Of the topics on this list, the relevant knowledge of probability is not something that would be a prerequisite for third-semester calculus at the school where I taught before I retired. You say that you covered it during the your own class, but one shouldn't expect students to master topics like this that, from your description, are more like a preview of later coursework.

(3) The English is not very good and requires a great deal of additional effort to parse.

You describe the class as an honors class in which most of the students are math education majors. This sounds like a little bit of a jackalope. Math education majors are typically extremely poor math students. Perhaps the honors designation led you to overestimate the level of the students. At my school, "honors" is extremely watered down and pretty meaningless.

I would expect students' response to this question to depend on factors such as how much partial credit you gave, how difficult the other questions on the test were, and what grading scale you're using.

• Thank you for your comments. It is indeed a long solution. The one I typed (because we always go over the solutions in great detail) is 3 pages long and I also wrote an additional solution (alternative) that is about 1 page. Yes, the Probability was of course a preview. I normally (and this is always an issue for me), I always get told that I go way above and beyond the syllabus. Normally the admin. likes it, but sometimes they interpret it as overkill. Depends who you ask. Yes, I found out quite a bit late in the game that math ed majors are bad at Math.
– Wasp
Jun 8 '21 at 23:59
• One of the reasons why I am very demanding of them is because 1) it is an honors class and even though you are correct about the meaningless part these days, I personally take it very seriously and 2) My own personal philosophy is that I believe it should be a requirement for math educators to be Math majors and be pure mathematicians. This is a method widely used in many countries around the world and I think we could benefit from it here. But, I do very much accept and acknowledge your critique.
– Wasp
Jun 9 '21 at 0:01
• You describe the class as an honors class in which most of the students are math education majors. --- I was thinking of commenting about this yesterday (maybe the day before yesterday) and wound up not taking the time to do so. To me (but maybe this is not current course name usage anymore), honors calculus would be a class for exceptional students. I once graded for a year-long honors multivariable calculus class (covered the entire Volume 2 of Apostol's 2-volume calculus text), (continued) Jun 9 '21 at 14:15
• total enrollment of 6 or 7 students, and 2 went on to be math Ph.D's (one a Princeton Ph.D, the other wound up being a full professor at Univ. of Arizona), one is a reasonably respected medical researcher, and I strongly suspect ALL attended graduate school in a science field. At last half of the students took the Putnam exam more than once, and one was a top 100 scorer (more than once, I think). However, at this same university, there were two tracks for upper level math courses (abstract algebra, linear algebra, real analysis, etc.) -- one for math majors and the other for math ed majors. Jun 9 '21 at 14:15