# KarlRosaen

More probability HW, making sense of the odds ratio underlying logistic regression

## Homework

This morning I solved problem 7 from All of Statistics. It was tough for me, taking about an hour and a half, and I needed to peek at the solution to get over one minor hump, but I was proud to have nearly gotten it, including thinking to use induction to prove the 2nd part on my own.

The part that got me was this step:

$(A_{n+1} \cap (\bigcup_{i=1}^{n} A_i)^c) \cup \bigcup_{i=1}^{n} A_i$
= $[\bigcup_{i=1}^{n} A_i \cup A_{n+1}] \cap [\bigcup_{i=1}^{n} A_i \cup (\bigcup_{i=1}^{n} A_i)^c]$

because I didn't remember / know that union distributes over intersection in set theory! I won't forget it now that I spent 20 minutes stumped on it :)

It's also interesting that this problem uses a similar trick to problem 1 in crafting an intermediate variable that can be proven to be disjoint so that you can add up the probabilities.

## Back to Python ML

### Making sense of the odds ratio

In the Python ML book it goes through how the cost function is derived for logistic regression using the odds ratio, which is $\frac{p}{(1 - p)}$. I was left unsatisfied with this being presented as matter of fact so went googling around to get a bit more of a sense of why this ratio would be used instead of the direct probability.