Proportional density function question

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.

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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.

Answer 1

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 $\pi/2$.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.