HW0 - Did some work, and added unanswered questions. Homework 0 is incomplete

main.aux

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main.fdb_latexmk

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 # Fdb version 4
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main.fls

@@ -7,20 +7,16 @@ INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\base\article.cls
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 INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\tfm\jknappen\ec\tcrm1000.tfm
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main.log

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+]
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main.tex

@@ -1,5 +1,5 @@
 \documentclass[titlepage]{article}
-\usepackage{graphicx} % Required for inserting images
+\usepackage{amsmath}
 
 \setcounter{section}{-1}
 
@@ -26,7 +26,7 @@ When it's time to submit an assignment, I will export this document as a PDF fil
 \newpage
 
 \section{Homework 0}
-\textit{By turning in this assignment, I agree to abide by SFSU’s academic integrity code and declare that all of my solutions are my own work: }
+\textit{By turning in this assignment, I agree to abide by SFSU's academic integrity code and declare that all of my solutions are my own work: }
 
 \subsection{About You}
 \begin{enumerate}
@@ -42,18 +42,59 @@ When it's time to submit an assignment, I will export this document as a PDF fil
         \item Data Structures
     \end{itemize}
     This list consists of the courses I think are relevant to this class; I've taken other CS courses, of course.
-    \item Yes, I am okay with being called on. I won't always know the answer, and I might embarrass myself sometimes, but that's alright because I believe it will force me to pay more attention and learn better, and as a result be more prepared for assignments and exams.
+    \item [c.] Yes, I am okay with being called on. I won't always know the answer, and I might embarrass myself sometimes, but that's alright because I believe it will force me to pay more attention and learn better, and as a result be more prepared for assignments and exams.
 \end{enumerate}
 
 \subsection{Optimization}
 \begin{itemize}
     \item [a.] Let a, b, and c be positive real numbers. Consider the quadratic function
-    $$
+        $$f(\theta) = a{\theta}^2 + b\theta + c$$
-    f(\theta) = a{\theta}^2 + b\theta + c
+        Note that $\theta$ here is a real number. What value of $\theta$ minimizes $f(\theta)$?\\[0.1in]
-    $$
+        Solution:
-    Note that $\theta$ here is a real number. What value of $\theta$ minimizes $f(\theta)$?
+        Since this is a positive parabola, we can use the formula for the vertex of a parabola, $\frac{-b}{2a}$, to find the minimum:\\
-    Solution:
+        $$
+        \boxed{\theta = \frac{-b}{2a}}
+        $$
     
+    \item [b.] Let $x_1,\dots,x_n$ be real numbers. Consider the quadratic function
+        $$g(\theta) = \sum^n_{i=1}(\theta - x_i)^2.$$
+        What value of $\theta$ minimizes $g(\theta)$?
+
+        {\LARGE{TODO}}
+
+    \item [c.] Let $x_1,\dots,x_n$ again be real numbers, and let $w_1,\dots,w_n$ be positive real numbers that we can interpret as representing the importance of each of the $x_i$'s. Consider the weighted quadratic function
+        $$h(\theta) = \sum_{i=1}^{n}w_i(\theta - x_i)^2.$$
+        What value of $\theta$ minimizes $h(\theta)$?
+
+    \item [d.] What issue could arise in the minimization of $h$ if some of the $w_i$'s are negative?
+\end{itemize}
+
+\subsection{Probability}
+
+\begin{itemize}
+    \item [a.] Consider a standard 52-card deck of cards with 13 card values (Ace, King, Queen, Jack, and
+    2-10) in each of the four suits (clubs, diamonds, hearts, spades). If a card is drawn at random, what
+    is the probability that it is a spade or a two?
+
+    \item [b.] Two factories — Factory A and Factory B — design batteries to be used in mobile phones.
+    Factory A produces 60\% of all batteries, and Factory B produces the other 40\%. 2\% of Factory A's
+    batteries have defects, and 4\% of Factory B's batteries have defects. What is the probability that a
+    battery is both made by Factory A and defective?
+
+    \item [c.] Consider the following (made up) facts about COVID incidence and testing:
+        \begin{itemize}
+            \item In the absence of any special information, the probability that a person has COVID is 1\%.
+            \item If a person has COVID, the probability that a test will correctly read positive is 80\%.
+            \item If a person does not have COVID, the probability that a test will incorrectly produce a false positive is 10\%.
+        \end{itemize}
+
+        Suppose you take a COVID test and it reads positive. Given the facts above, what is the probability
+that you have COVID?
+
+    \item [d. ] Suppose you repeatedly roll a fair six-sided die until you roll a 1 (and then you stop). Every
+    time you roll a 3, you win a points, and every time you roll a 6, you lose b points. You do not win or
+    lose any points if you roll a 2, 4, or 5. What is the expected number of points (as a function of a and
+    b) you will have when you stop?
 \end{itemize}
 
 \end{document}

titlePage.aux

@@ -16,4 +16,5 @@
 \setcounter{subparagraph}{0}
 \setcounter{figure}{0}
 \setcounter{table}{0}
+\setcounter{parentequation}{0}
 }