Problem Set 0

Due Friday September 5 at 5PM

Nature laughs at the difficulties of integration.

This is ostensibly calculus review, but each problem is a piece of probability in disguise. We will illuminate these connections throughout the semester, and I will refer to Problem Set 0 often. Stay tuned!

Problem 0

Recommend some music for us to listen to while we grade this.

Problem 1

Explain why this is horrific notation:

\[ \int_0^x f(x)\,\textrm{d} x. \]

How should it be fixed?

Problem 2

Simplify this:

\[ \ln\left(e^{a_1}e^{a_2}e^{a_3}\cdots e^{a_n}\right). \]

Problem 3

Assume \(\lambda>0\) is a constant and compute

\[ \sum\limits_{n=0}^\infty n \frac{\lambda^n}{n!}e^{-\lambda} . \]

Problem 4

Here is a very silly function:

\[ h(x) = \exp\left(-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}\right) ,\quad -\infty<x<\infty . \]

Treat \(-\infty<\mu<\infty\) and \(\sigma>0\) as constants and compute the value(s) of \(x\) at which \(h\) has inflection points.

Here is an example of what \(h\) might look like in the special case where \(\mu = 1\) and \(\sigma=2\):

Code
m = 1
s = 2
par(mar = c(4, 4, 0.1, 4))
curve(exp(-0.5 * ((x - m) / s)^2), 
      from = -6, to = 8, n = 500,
      xlab = "x", ylab = "h(x)",
      xaxt = "n")
axis(1, at = -6:8)

Before you start doing any math, can you use the picture to guess what the answer will be?

Problem 5

Here is another inordinately silly function:

\[ \Gamma(x)=\int_0^\infty y^{x-1}e^{-y}\,\textrm{d} y,\quad x>0. \]

Prove that \(\Gamma(x+1)=x\Gamma(x)\).

Start on the left-hand side by writing out \(\Gamma(x+1)\) and evaluating the integral by parts.

Problem 6

Let \(f\) be any function with the following properties:

  • \(f\) is twice continuously differentiable in a neighborhood of zero1;
  • \(f(0) = 0\);
  • \(f'(0) = 0\);
  • \(f''(0) = 1\).

Assume \(t\) is a constant and compute

\[ \lim_{x\to\infty} xf\left(\frac{t}{\sqrt{x}}\right) . \]

Continuity?

A full credit solution must clearly explain how and why continuity is being used along the way.

Problem 7

Consider this integral:

\[ \int_2^\infty \frac{1}{x(\ln x)^p}\textrm{d} x . \]

  1. Use R to create a single plot with many lines, each graphing the integrand for a different value of \(p\). Consider \(p\) equal to -2, -1.5, -1, 0, 1, and 5, and make the \(x\)-axis of your plot run from 2 to 15.
  2. Show that \(\lim_{x\to\infty}\frac{1}{x(\ln x)^p}=0\) for all values of \(-\infty<p<\infty\).
  3. For what values of \(p\) does the integral converge? When it does converge, what is its value?
  4. Consult the picture you created in part (a), and write a few sentences explaining conceptually why the integral converges for some values of \(p\) but not others.

When taking the limit or evaluating the integral, can you use the same technique for all values of \(p\), or do you need a different technique depending on what \(p\) is?

Submission

You are free to compose your solutions for this problem set however you wish (scan or photograph written work, handwriting capture on a tablet device, LaTeX, Quarto, whatever) as long as the final product is a single PDF file. You must upload this to Gradescope and mark the pages associated with each problem.

Do not forget to include the following:

  • For each problem, please acknowledge your collaborators;
  • If a problem required you to code something, please include both the code and the output. “Including the code” can be as crude as a screenshot, but you might also use Quarto to get a nice lil’ pdf that you can merge with the rest of your submission.

Footnotes

  1. This means that \(f\) and its first two derivatives are all continuous functions at and around zero.↩︎