Example writeup

Example Writeup

Homeworks should be written up clearly and succinctly; you may lose points if your answers are unclear or unnecessarily complicated. This is an example of what we are looking for.

Simple Question

Question: In big-O notation, how many contiguous subsequences are there of a list of $n$ numbers?

Answer: There are $O(n)$ choices for the starting position of the subsequence and $O(n)$ choices for the ending position. Therefore, there are $O(n^2)$ possible contiguous subsequences.

Here, using words is precise enough and easy to read.

More Involved Question

Question: Suppose you flip a sequence of independent fair coins until you get heads, and on each preceding tails flip, you roll a 6-sided fair dice. What is the expected value of the sum of the die?

Answer:

Let $Z_1, Z_2, \dots$ be the values of the coin tosses.
Let $X_1, X_2, \dots$ be the values of the dice rolls (let $X_i=0$ if the $i$th dice was not rolled).
Let $A_i$ be the event that $i$th dice was rolled, which is exactly when $Z_1, \dots, Z_i$ are all tails. This happens with probability $\mathbb{P}(A_i) = 2^{-i}$ by independence of the coin flips.
If the $i$th dice is rolled, then expected contribution to the sum is $\mathbb{E}[X_i|A_i]=3.5$; otherwise, it is zero.
Finally, the expected value of the sum, by linearity of expectation, is
$\sum_{i=1}^{\infty}\mathbb{E}[X_i] = \sum_{i=1}^{\infty}(3.5\cdot 2^{-i}+0) = 3.5$

General guidelines:

Try to balance mathematical notation and words.
Introduce notation only if you need to refer back to it later in an exact way and words aren't enough.
If you introduce your own notation in the problem, define everything up front.
When you make a statement, it should be clear whether it is assumed from the problem, or derived from a previous step, or something that you're trying to derive.
Regardless of whether you use mathematical notation of words, be clear and precise.
You can break down a solution using bullet points to make things more readable.