Assignment 1
Assigned 2024-08-27, due 2024-09-03
Question 1
-
Find the link for the discussion board on the class website. Join it, and write a comment with a random Math/CS fact. (Include a screenshot with your homework.)
-
Find the link for the discord server on the class website post a random Math/CS fact. (Include a screenshot with your homework.)
Question 2
Follow the instructions at the start of this notebook (download) and get yourself setup with Python, and run the notebook. You have two choices:
-
Install Python and the required libraries on your machine (much faster, but some setup required). Instructions are in the notebook.
-
Import the notebook into Google Colab (slow, but no setup required.)
Either way, ensure the notebook runs without any error messages. (We will walk through what it does in class, so don’t worry about understanding the content just yet.)
Include a screenshot showing you managed to run the notebook with your homework. (Your screenshot only needs to show the last few cells completing; no need to show everything.)
Question 3
Prove Proposition 5 (Box Mueller).
Question 4
Suppose $U$, $V$ are independent $\Unif( (0, 1) )$ distributed random variables. Find a transformation $T$ such that $T(U, V)$ is uniformly distributed on the unit ball $B(0, 1) \subseteq \R^2$. (You should also check that $T(U, V)$ indeed has the right distribution.)
Note: The unit ball $B(0, 1) \subseteq \R^2$ is defined to be the set of all $x \in \R^2$ such that $\abs{x} < 1$. Also $\abs{x} = (\sum x_i^2)^{1/2}$ is the length of the vector $x$.