Problem Set 5

Due Monday October 27 at 3PM

Brace yourself for a gear shift. Now, seemingly out of nowhere, STA 240 becomes a calculus class.

Problem 0

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

Problem 1

From now to the end of the course, we will use all of the tools of single-variable calculus. You got a preview of this on Problem Set 0, and our course page also includes some math refreshers (derivatives, integrals, etc). Please review these.

What areas of calculus are you the least comfortable with at the moment? How do you think the course materials could be improved to better prepare you for this aspect of the course?

Problem 2

Consider this function:

\[ F(x) = \begin{cases} 1 - e^{-g(x)} & x\geq 0\\ 0 & \text{else}. \end{cases} \]

Assume \(g(x)\) is continuous. I want \(F(x)\) to be a continuous function, and I want \(F(x)\) to be a valid cumulative distribution function (CDF). What properties must \(g(x)\) satisfy in order to make that happen? Give three examples of \(g\) that have these properties.

Problem 3

An absolutely continuous random variable \(X\) has pdf

\[ f(x)=\begin{cases} \frac{3}{22}[5 - (x-1)^2] & 1\leq x \leq3\\ 0 & \text{else}. \end{cases} \]

  1. What is the range of \(X\)?
  2. Confirm that \(f\) is a valid pdf.
  3. Derive the formula for the cdf of \(X\) and plot it.
  4. Compute \(P(0.9 < X < 1.1)\).
  5. Compute \(E(X)\).
  6. Compute \(\text{var}(X)\).

Problem 4

Recall the gamma function

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

Show that \(\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\).

Problem 5

A point is chosen at random on a line segment of length \(L\). Interpret this statement, and find the probability that the ratio of the shorter to the longer segment is less than \(\frac{1}{4}\).

It might help to read about the continuous uniform distribution.

Problem 6

Here is the cdf of an absolutely continuous random variable \(X\):

\[ F(x) = \begin{cases} 1-\exp\left(-\sqrt{x}\right)&x\geq 0\\ 0&\text{else}. \end{cases} \]

  1. What is \(P(4\leq X < 9)\)?
  2. What is the pdf of \(X\)?
  3. Compute \(E(X^n)\) for any \(n\in\mathbb{N}\).
  4. What is the variance of \(X\)?
  5. What is the median of \(X\)?
Hint

All of the moments of this distribution are finite, but nevertheless, it doesn’t have a moment-generating function (MGF). So you’ll hit a dead-end if you try using that to compute the moments. Instead, use LOTUS. It’s a subtle calculation, but it actually works out pretty nice.

Problem 7

One of the technical themes of our course is “how do I compute an expected value without wanting to vomit?” We will have many tools in our tool belt for getting this done, and you must become fluent in knowing what to use when. Below are two nifty facts that might come in handy one day.

Let \(X\) be an absolutely continuous random variable with pdf \(f\) and cdf \(F\).

  1. Show that:

\[ E(X)=\int_0^1F^{-1}(x)\,\text{d} x. \]

  1. If we make the extra assumption that \(X\) is a non-negative random variable, show that:

\[ E(X)=\int_0^\infty P(X>x)\,\text{d} x. \]

There are at least two ways to do part b. If you’ve taken multivariable calculus, you can use double integral tricks. If you haven’t taken multivariable calculus yet, try integrating by parts.

Problem 8

Let \(X\) be any absolutely continuous random variable with pdf \(f\) and cdf \(F\), and assume that \(E[(X-a)^2]\) and \(E\left[|X-a|\right]\) are finite for all \(a\in \mathbb{R}\).

  1. Compute and interpret

\[ a_0=\underset{a\in\mathbb{R}}{\arg\min}\,E[(X-a)^2]. \]

  1. Compute and interpret

\[ b_0=\underset{b\in\mathbb{R}}{\arg\min}\,E\left[|X-b|\right]. \]

Problem 9

  1. Compute the moment generating function of the geometric distribution. For what values of \(t\) is it defined?
  2. Use the mgf to compute the mean and variance of the geometric distribution.

Problem 10

Consider \(X\sim\textrm{Gamma}(\alpha,\,\beta)\).

  1. Find \(M_X(t)=E[e^{tX}]\), the moment generating function of \(X\). For what values of \(t\) is it defined?
  2. Use the moment-generating function to compute \(E(X)\).
  3. Use the moment-generating function to compute \(\text{var}(X)\).
  4. If \(c>0\), what is the distribution of \(Y=cX\)?

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.