diff --git a/.gitignore b/.gitignore index e69de29..faae5ab 100644 --- a/.gitignore +++ b/.gitignore @@ -0,0 +1,6 @@ +**/*.aux +**/*.fdb_latexmk +**/*.fls +**/*.log +**/*.synctex.gz + diff --git a/general-template/homework.tex b/general-template/homework.tex new file mode 100644 index 0000000..872e6f5 --- /dev/null +++ b/general-template/homework.tex @@ -0,0 +1,539 @@ +\documentclass{article} + +\usepackage{fancyhdr} +\usepackage{extramarks} +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{amsfonts} +\usepackage{tikz} +\usepackage[plain]{algorithm} +\usepackage{algpseudocode} + +\usetikzlibrary{automata,positioning} + +% +% Basic Document Settings +% + +\topmargin=-0.45in +\evensidemargin=0in +\oddsidemargin=0in +\textwidth=6.5in +\textheight=9.0in +\headsep=0.25in + +\linespread{1.1} + +\pagestyle{fancy} +\lhead{\hmwkAuthorName} +\chead{\hmwkClass\ (\hmwkClassInstructor\ \hmwkClassTime): \hmwkTitle} +\rhead{\firstxmark} +\lfoot{\lastxmark} +\cfoot{\thepage} + +\renewcommand\headrulewidth{0.4pt} +\renewcommand\footrulewidth{0.4pt} + +\setlength\parindent{0pt} + +% +% Create Problem Sections +% + +\newcommand{\enterProblemHeader}[1]{ + \nobreak\extramarks{}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} + \nobreak\extramarks{Problem \arabic{#1} (continued)}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} +} + +\newcommand{\exitProblemHeader}[1]{ + \nobreak\extramarks{Problem \arabic{#1} (continued)}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} + \stepcounter{#1} + \nobreak\extramarks{Problem \arabic{#1}}{}\nobreak{} +} + +\setcounter{secnumdepth}{0} +\newcounter{partCounter} +\newcounter{homeworkProblemCounter} +\setcounter{homeworkProblemCounter}{1} +\nobreak\extramarks{Problem \arabic{homeworkProblemCounter}}{}\nobreak{} + +% +% Homework Problem Environment +% +% This environment takes an optional argument. When given, it will adjust the +% problem counter. This is useful for when the problems given for your +% assignment aren't sequential. See the last 3 problems of this template for an +% example. +% +\newenvironment{homeworkProblem}[1][-1]{ + \ifnum#1>0 + \setcounter{homeworkProblemCounter}{#1} + \fi + \section{Problem \arabic{homeworkProblemCounter}} + \setcounter{partCounter}{1} + \enterProblemHeader{homeworkProblemCounter} +}{ + \exitProblemHeader{homeworkProblemCounter} +} + +% +% Homework Details +% - Title +% - Due date +% - Class +% - Section/Time +% - Instructor +% - Author +% + +\newcommand{\hmwkTitle}{Homework\ \#2} +\newcommand{\hmwkDueDate}{February 12, 2014} +\newcommand{\hmwkClass}{Calculus} +\newcommand{\hmwkClassTime}{Section A} +\newcommand{\hmwkClassInstructor}{Professor Isaac Newton} +\newcommand{\hmwkAuthorName}{\textbf{Josh Davis} \and \textbf{Davis Josh}} + +% +% Title Page +% + +\title{ + \vspace{2in} + \textmd{\textbf{\hmwkClass:\ \hmwkTitle}}\\ + \normalsize\vspace{0.1in}\small{Due\ on\ \hmwkDueDate\ at 3:10pm}\\ + \vspace{0.1in}\large{\textit{\hmwkClassInstructor\ \hmwkClassTime}} + \vspace{3in} +} + +\author{\hmwkAuthorName} +\date{} + +\renewcommand{\part}[1]{\textbf{\large Part \Alph{partCounter}}\stepcounter{partCounter}\\} + +% +% Various Helper Commands +% + +% Useful for algorithms +\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}} + +% For derivatives +\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)} + +% For partial derivatives +\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)} + +% Integral dx +\newcommand{\dx}{\mathrm{d}x} + +% Alias for the Solution section header +\newcommand{\solution}{\textbf{\large Solution}} + +% Probability commands: Expectation, Variance, Covariance, Bias +\newcommand{\E}{\mathrm{E}} +\newcommand{\Var}{\mathrm{Var}} +\newcommand{\Cov}{\mathrm{Cov}} +\newcommand{\Bias}{\mathrm{Bias}} + +\begin{document} + +\maketitle + +\pagebreak + +\begin{homeworkProblem} + Give an appropriate positive constant \(c\) such that \(f(n) \leq c \cdot + g(n)\) for all \(n > 1\). + + \begin{enumerate} + \item \(f(n) = n^2 + n + 1\), \(g(n) = 2n^3\) + \item \(f(n) = n\sqrt{n} + n^2\), \(g(n) = n^2\) + \item \(f(n) = n^2 - n + 1\), \(g(n) = n^2 / 2\) + \end{enumerate} + + \textbf{Solution} + + We solve each solution algebraically to determine a possible constant + \(c\). + \\ + + \textbf{Part One} + + \[ + \begin{split} + n^2 + n + 1 &= + \\ + &\leq n^2 + n^2 + n^2 + \\ + &= 3n^2 + \\ + &\leq c \cdot 2n^3 + \end{split} + \] + + Thus a valid \(c\) could be when \(c = 2\). + \\ + + \textbf{Part Two} + + \[ + \begin{split} + n^2 + n\sqrt{n} &= + \\ + &= n^2 + n^{3/2} + \\ + &\leq n^2 + n^{4/2} + \\ + &= n^2 + n^2 + \\ + &= 2n^2 + \\ + &\leq c \cdot n^2 + \end{split} + \] + + Thus a valid \(c\) is \(c = 2\). + \\ + + \textbf{Part Three} + + \[ + \begin{split} + n^2 - n + 1 &= + \\ + &\leq n^2 + \\ + &\leq c \cdot n^2/2 + \end{split} + \] + + Thus a valid \(c\) is \(c = 2\). + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Let \(\Sigma = \{0, 1\}\). Construct a DFA \(A\) that recognizes the + language that consists of all binary numbers that can be divided by 5. + \\ + + Let the state \(q_k\) indicate the remainder of \(k\) divided by 5. For + example, the remainder of 2 would correlate to state \(q_2\) because \(7 + \mod 5 = 2\). + + \begin{figure}[h] + \centering + \begin{tikzpicture}[shorten >=1pt,node distance=2cm,on grid,auto] + \node[state, accepting, initial] (q_0) {$q_0$}; + \node[state] (q_1) [right=of q_0] {$q_1$}; + \node[state] (q_2) [right=of q_1] {$q_2$}; + \node[state] (q_3) [right=of q_2] {$q_3$}; + \node[state] (q_4) [right=of q_3] {$q_4$}; + \path[->] + (q_0) + edge [loop above] node {0} (q_0) + edge node {1} (q_1) + (q_1) + edge node {0} (q_2) + edge [bend right=-30] node {1} (q_3) + (q_2) + edge [bend left] node {1} (q_0) + edge [bend right=-30] node {0} (q_4) + (q_3) + edge node {1} (q_2) + edge [bend left] node {0} (q_1) + (q_4) + edge node {0} (q_3) + edge [loop below] node {1} (q_4); + \end{tikzpicture} + \caption{DFA, \(A\), this is really beautiful, ya know?} + \label{fig:multiple5} + \end{figure} + + \textbf{Justification} + \\ + + Take a given binary number, \(x\). Since there are only two inputs to our + state machine, \(x\) can either become \(x0\) or \(x1\). When a 0 comes + into the state machine, it is the same as taking the binary number and + multiplying it by two. When a 1 comes into the machine, it is the same as + multipying by two and adding one. + \\ + + Using this knowledge, we can construct a transition table that tell us + where to go: + + \begin{table}[ht] + \centering + \begin{tabular}{c || c | c | c | c | c} + & \(x \mod 5 = 0\) + & \(x \mod 5 = 1\) + & \(x \mod 5 = 2\) + & \(x \mod 5 = 3\) + & \(x \mod 5 = 4\) + \\ + \hline + \(x0\) & 0 & 2 & 4 & 1 & 3 \\ + \(x1\) & 1 & 3 & 0 & 2 & 4 \\ + \end{tabular} + \end{table} + + Therefore on state \(q_0\) or (\(x \mod 5 = 0\)), a transition line should + go to state \(q_0\) for the input 0 and a line should go to state \(q_1\) + for input 1. Continuing this gives us the Figure~\ref{fig:multiple5}. +\end{homeworkProblem} + +\begin{homeworkProblem} + Write part of \alg{Quick-Sort($list, start, end$)} + + \begin{algorithm}[] + \begin{algorithmic}[1] + \Function{Quick-Sort}{$list, start, end$} + \If{$start \geq end$} + \State{} \Return{} + \EndIf{} + \State{} $mid \gets \Call{Partition}{list, start, end}$ + \State{} \Call{Quick-Sort}{$list, start, mid - 1$} + \State{} \Call{Quick-Sort}{$list, mid + 1, end$} + \EndFunction{} + \end{algorithmic} + \caption{Start of QuickSort} + \end{algorithm} +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Suppose we would like to fit a straight line through the origin, i.e., + \(Y_i = \beta_1 x_i + e_i\) with \(i = 1, \ldots, n\), \(\E [e_i] = 0\), + and \(\Var [e_i] = \sigma^2_e\) and \(\Cov[e_i, e_j] = 0, \forall i \neq + j\). + \\ + + \part + + Find the least squares esimator for \(\hat{\beta_1}\) for the slope + \(\beta_1\). + \\ + + \solution + + To find the least squares estimator, we should minimize our Residual Sum + of Squares, RSS: + + \[ + \begin{split} + RSS &= \sum_{i = 1}^{n} {(Y_i - \hat{Y_i})}^2 + \\ + &= \sum_{i = 1}^{n} {(Y_i - \hat{\beta_1} x_i)}^2 + \end{split} + \] + + By taking the partial derivative in respect to \(\hat{\beta_1}\), we get: + + \[ + \pderiv{ + \hat{\beta_1} + }{RSS} + = -2 \sum_{i = 1}^{n} {x_i (Y_i - \hat{\beta_1} x_i)} + = 0 + \] + + This gives us: + + \[ + \begin{split} + \sum_{i = 1}^{n} {x_i (Y_i - \hat{\beta_1} x_i)} + &= \sum_{i = 1}^{n} {x_i Y_i} - \sum_{i = 1}^{n} \hat{\beta_1} x_i^2 + \\ + &= \sum_{i = 1}^{n} {x_i Y_i} - \hat{\beta_1}\sum_{i = 1}^{n} x_i^2 + \end{split} + \] + + Solving for \(\hat{\beta_1}\) gives the final estimator for \(\beta_1\): + + \[ + \begin{split} + \hat{\beta_1} + &= \frac{ + \sum {x_i Y_i} + }{ + \sum x_i^2 + } + \end{split} + \] + + \pagebreak + + \part + + Calculate the bias and the variance for the estimated slope + \(\hat{\beta_1}\). + \\ + + \solution + + For the bias, we need to calculate the expected value + \(\E[\hat{\beta_1}]\): + + \[ + \begin{split} + \E[\hat{\beta_1}] + &= \E \left[ \frac{ + \sum {x_i Y_i} + }{ + \sum x_i^2 + }\right] + \\ + &= \frac{ + \sum {x_i \E[Y_i]} + }{ + \sum x_i^2 + } + \\ + &= \frac{ + \sum {x_i (\beta_1 x_i)} + }{ + \sum x_i^2 + } + \\ + &= \frac{ + \sum {x_i^2 \beta_1} + }{ + \sum x_i^2 + } + \\ + &= \beta_1 \frac{ + \sum {x_i^2 \beta_1} + }{ + \sum x_i^2 + } + \\ + &= \beta_1 + \end{split} + \] + + Thus since our estimator's expected value is \(\beta_1\), we can conclude + that the bias of our estimator is 0. + \\ + + For the variance: + + \[ + \begin{split} + \Var[\hat{\beta_1}] + &= \Var \left[ \frac{ + \sum {x_i Y_i} + }{ + \sum x_i^2 + }\right] + \\ + &= + \frac{ + \sum {x_i^2} + }{ + \sum x_i^2 \sum x_i^2 + } \Var[Y_i] + \\ + &= + \frac{ + \sum {x_i^2} + }{ + \sum x_i^2 \sum x_i^2 + } \Var[Y_i] + \\ + &= + \frac{ + 1 + }{ + \sum x_i^2 + } \Var[Y_i] + \\ + &= + \frac{ + 1 + }{ + \sum x_i^2 + } \sigma^2 + \\ + &= + \frac{ + \sigma^2 + }{ + \sum x_i^2 + } + \end{split} + \] + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Prove a polynomial of degree \(k\), \(a_kn^k + a_{k - 1}n^{k - 1} + \hdots + + a_1n^1 + a_0n^0\) is a member of \(\Theta(n^k)\) where \(a_k \hdots a_0\) + are nonnegative constants. + + \begin{proof} + To prove that \(a_kn^k + a_{k - 1}n^{k - 1} + \hdots + a_1n^1 + + a_0n^0\), we must show the following: + + \[ + \exists c_1 \exists c_2 \forall n \geq n_0,\ {c_1 \cdot g(n) \leq + f(n) \leq c_2 \cdot g(n)} + \] + + For the first inequality, it is easy to see that it holds because no + matter what the constants are, \(n^k \leq a_kn^k + a_{k - 1}n^{k - 1} + + \hdots + a_1n^1 + a_0n^0\) even if \(c_1 = 1\) and \(n_0 = 1\). This + is because \(n^k \leq c_1 \cdot a_kn^k\) for any nonnegative constant, + \(c_1\) and \(a_k\). + \\ + + Taking the second inequality, we prove it in the following way. + By summation, \(\sum\limits_{i=0}^k a_i\) will give us a new constant, + \(A\). By taking this value of \(A\), we can then do the following: + + \[ + \begin{split} + a_kn^k + a_{k - 1}n^{k - 1} + \hdots + a_1n^1 + a_0n^0 &= + \\ + &\leq (a_k + a_{k - 1} \hdots a_1 + a_0) \cdot n^k + \\ + &= A \cdot n^k + \\ + &\leq c_2 \cdot n^k + \end{split} + \] + + where \(n_0 = 1\) and \(c_2 = A\). \(c_2\) is just a constant. Thus the + proof is complete. + \end{proof} +\end{homeworkProblem} + +\pagebreak + +% +% Non sequential homework problems +% + +% Jump to problem 18 +\begin{homeworkProblem}[18] + Evaluate \(\sum_{k=1}^{5} k^2\) and \(\sum_{k=1}^{5} (k - 1)^2\). +\end{homeworkProblem} + +% Continue counting to 19 +\begin{homeworkProblem} + Find the derivative of \(f(x) = x^4 + 3x^2 - 2\) +\end{homeworkProblem} + +% Go back to where we left off +\begin{homeworkProblem}[6] + Evaluate the integrals + \(\int_0^1 (1 - x^2) \dx\) + and + \(\int_1^{\infty} \frac{1}{x^2} \dx\). +\end{homeworkProblem} + +\end{document} diff --git a/labs-set-1/Lab 2 - 13.1 Apps.pdf b/labs-set-1/Lab 2 - 13.1 Apps.pdf new file mode 100644 index 0000000..1f20708 Binary files /dev/null and b/labs-set-1/Lab 2 - 13.1 Apps.pdf differ diff --git a/labs-set-1/Lab 3 - 13.2.pdf b/labs-set-1/Lab 3 - 13.2.pdf new file mode 100644 index 0000000..f685fb6 --- /dev/null +++ b/labs-set-1/Lab 3 - 13.2.pdf @@ -0,0 +1,29476 @@ +%PDF-1.4 +%âãÏÓ +3 0 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b/labs-set-1/homework.pdf differ diff --git a/labs-set-1/homework.tex b/labs-set-1/homework.tex new file mode 100644 index 0000000..c8823d2 --- /dev/null +++ b/labs-set-1/homework.tex @@ -0,0 +1,541 @@ +\documentclass{article} + +\usepackage{fancyhdr} +\usepackage{extramarks} +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{amsfonts} +\usepackage{tikz} +\usepackage[plain]{algorithm} +\usepackage{algpseudocode} +\usepackage{comment} +\usetikzlibrary{automata,positioning} + +% +% Basic Document Settings +% + +\topmargin=-0.45in +\evensidemargin=0in +\oddsidemargin=0in +\textwidth=6.5in +\textheight=9.0in +\headsep=0.25in + +\linespread{1.1} + +\pagestyle{fancy} +\lhead{\hmwkAuthorName} +\chead{\hmwkClass\ (\hmwkClassInstructor\ \hmwkClassTime): \hmwkTitle} +\rhead{\firstxmark} +\lfoot{\lastxmark} +\cfoot{\thepage} + +\renewcommand\headrulewidth{0.4pt} +\renewcommand\footrulewidth{0.4pt} + +\setlength\parindent{0pt} + +% +% Create Problem Sections +% + +\newcommand{\enterProblemHeader}[1]{ + \nobreak\extramarks{}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} + \nobreak\extramarks{Problem \arabic{#1} (continued)}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} +} + +\newcommand{\exitProblemHeader}[1]{ + \nobreak\extramarks{Problem \arabic{#1} (continued)}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} + \stepcounter{#1} + \nobreak\extramarks{Problem \arabic{#1}}{}\nobreak{} +} + +\setcounter{secnumdepth}{0} +\newcounter{partCounter} +\newcounter{homeworkProblemCounter} +\setcounter{homeworkProblemCounter}{1} +\nobreak\extramarks{Problem \arabic{homeworkProblemCounter}}{}\nobreak{} + +% +% Homework Problem Environment +% +% This environment takes an optional argument. When given, it will adjust the +% problem counter. This is useful for when the problems given for your +% assignment aren't sequential. See the last 3 problems of this template for an +% example. +% +\newenvironment{homeworkProblem}[1][-1]{ + \ifnum#1>0 + \setcounter{homeworkProblemCounter}{#1} + \fi + \section{Problem \arabic{homeworkProblemCounter}} + \setcounter{partCounter}{1} + \enterProblemHeader{homeworkProblemCounter} +}{ + \exitProblemHeader{homeworkProblemCounter} +} + +% +% Homework Details +% - Title +% - Due date +% - Class +% - Section/Time +% - Instructor +% - Author +% + +\newcommand{\hmwkTitle}{Homework\ \#1} +\newcommand{\hmwkDueDate}{September 21, 2025} +\newcommand{\hmwkClass}{Calc III} +\newcommand{\hmwkClassTime}{$6^{\text{th}}$ Hour}} +\newcommand{\hmwkClassInstructor}{Professor Foresee} +\newcommand{\hmwkAuthorName}{\textbf{Krishna A.}} + +% +% Title Page +% + +\title{ + \vspace{2in} + \textmd{\textbf{\hmwkClass:\ \hmwkTitle}}\\ + \normalsize\vspace{0.1in}\small{Due\ on\ \hmwkDueDate\ at 3:10pm}\\ + \vspace{0.1in}\large{\textit{\hmwkClassInstructor\ \hmwkClassTime}} + \vspace{3in} +} + +\author{\hmwkAuthorName} +\date{} + +\renewcommand{\part}[1]{\textbf{\large Part \Alph{partCounter}}\stepcounter{partCounter}\\} + +% +% Various Helper Commands +% + +% Useful for algorithms +\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}} + +% For derivatives +\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)} + +% For partial derivatives +\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)} + +% Integral dx +\newcommand{\dx}{\mathrm{d}x} + +% Alias for the Solution section header +\newcommand{\solution}{\textbf{\large Solution}} + +% Probability commands: Expectation, Variance, Covariance, Bias +\newcommand{\E}{\mathrm{E}} +\newcommand{\Var}{\mathrm{Var}} +\newcommand{\Cov}{\mathrm{Cov}} +\newcommand{\Bias}{\mathrm{Bias}} + +\begin{document} + +\maketitle + +\pagebreak + +\begin{comment} + +\begin{homeworkProblem} + Give an appropriate positive constant \(c\) such that \(f(n) \leq c \cdot + g(n)\) for all \(n > 1\). + + \begin{enumerate} + \item \(f(n) = n^2 + n + 1\), \(g(n) = 2n^3\) + \item \(f(n) = n\sqrt{n} + n^2\), \(g(n) = n^2\) + \item \(f(n) = n^2 - n + 1\), \(g(n) = n^2 / 2\) + \end{enumerate} + + \textbf{Solution} + + We solve each solution algebraically to determine a possible constant + \(c\). + \\ + + \textbf{Part One} + + \[ + \begin{split} + n^2 + n + 1 &= + \\ + &\leq n^2 + n^2 + n^2 + \\ + &= 3n^2 + \\ + &\leq c \cdot 2n^3 + \end{split} + \] + + Thus a valid \(c\) could be when \(c = 2\). + \\ + + \textbf{Part Two} + + \[ + \begin{split} + n^2 + n\sqrt{n} &= + \\ + &= n^2 + n^{3/2} + \\ + &\leq n^2 + n^{4/2} + \\ + &= n^2 + n^2 + \\ + &= 2n^2 + \\ + &\leq c \cdot n^2 + \end{split} + \] + + Thus a valid \(c\) is \(c = 2\). + \\ + + \textbf{Part Three} + + \[ + \begin{split} + n^2 - n + 1 &= + \\ + &\leq n^2 + \\ + &\leq c \cdot n^2/2 + \end{split} + \] + + Thus a valid \(c\) is \(c = 2\). + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Let \(\Sigma = \{0, 1\}\). Construct a DFA \(A\) that recognizes the + language that consists of all binary numbers that can be divided by 5. + \\ + + Let the state \(q_k\) indicate the remainder of \(k\) divided by 5. For + example, the remainder of 2 would correlate to state \(q_2\) because \(7 + \mod 5 = 2\). + + \begin{figure}[h] + \centering + \begin{tikzpicture}[shorten >=1pt,node distance=2cm,on grid,auto] + \node[state, accepting, initial] (q_0) {$q_0$}; + \node[state] (q_1) [right=of q_0] {$q_1$}; + \node[state] (q_2) [right=of q_1] {$q_2$}; + \node[state] (q_3) [right=of q_2] {$q_3$}; + \node[state] (q_4) [right=of q_3] {$q_4$}; + \path[->] + (q_0) + edge [loop above] node {0} (q_0) + edge node {1} (q_1) + (q_1) + edge node {0} (q_2) + edge [bend right=-30] node {1} (q_3) + (q_2) + edge [bend left] node {1} (q_0) + edge [bend right=-30] node {0} (q_4) + (q_3) + edge node {1} (q_2) + edge [bend left] node {0} (q_1) + (q_4) + edge node {0} (q_3) + edge [loop below] node {1} (q_4); + \end{tikzpicture} + \caption{DFA, \(A\), this is really beautiful, ya know?} + \label{fig:multiple5} + \end{figure} + + \textbf{Justification} + \\ + + Take a given binary number, \(x\). Since there are only two inputs to our + state machine, \(x\) can either become \(x0\) or \(x1\). When a 0 comes + into the state machine, it is the same as taking the binary number and + multiplying it by two. When a 1 comes into the machine, it is the same as + multipying by two and adding one. + \\ + + Using this knowledge, we can construct a transition table that tell us + where to go: + + \begin{table}[ht] + \centering + \begin{tabular}{c || c | c | c | c | c} + & \(x \mod 5 = 0\) + & \(x \mod 5 = 1\) + & \(x \mod 5 = 2\) + & \(x \mod 5 = 3\) + & \(x \mod 5 = 4\) + \\ + \hline + \(x0\) & 0 & 2 & 4 & 1 & 3 \\ + \(x1\) & 1 & 3 & 0 & 2 & 4 \\ + \end{tabular} + \end{table} + + Therefore on state \(q_0\) or (\(x \mod 5 = 0\)), a transition line should + go to state \(q_0\) for the input 0 and a line should go to state \(q_1\) + for input 1. Continuing this gives us the Figure~\ref{fig:multiple5}. +\end{homeworkProblem} + +\begin{homeworkProblem} + Write part of \alg{Quick-Sort($list, start, end$)} + + \begin{algorithm}[] + \begin{algorithmic}[1] + \Function{Quick-Sort}{$list, start, end$} + \If{$start \geq end$} + \State{} \Return{} + \EndIf{} + \State{} $mid \gets \Call{Partition}{list, start, end}$ + \State{} \Call{Quick-Sort}{$list, start, mid - 1$} + \State{} \Call{Quick-Sort}{$list, mid + 1, end$} + \EndFunction{} + \end{algorithmic} + \caption{Start of QuickSort} + \end{algorithm} +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Suppose we would like to fit a straight line through the origin, i.e., + \(Y_i = \beta_1 x_i + e_i\) with \(i = 1, \ldots, n\), \(\E [e_i] = 0\), + and \(\Var [e_i] = \sigma^2_e\) and \(\Cov[e_i, e_j] = 0, \forall i \neq + j\). + \\ + + \part + + Find the least squares esimator for \(\hat{\beta_1}\) for the slope + \(\beta_1\). + \\ + + \solution + + To find the least squares estimator, we should minimize our Residual Sum + of Squares, RSS: + + \[ + \begin{split} + RSS &= \sum_{i = 1}^{n} {(Y_i - \hat{Y_i})}^2 + \\ + &= \sum_{i = 1}^{n} {(Y_i - \hat{\beta_1} x_i)}^2 + \end{split} + \] + + By taking the partial derivative in respect to \(\hat{\beta_1}\), we get: + + \[ + \pderiv{ + \hat{\beta_1} + }{RSS} + = -2 \sum_{i = 1}^{n} {x_i (Y_i - \hat{\beta_1} x_i)} + = 0 + \] + + This gives us: + + \[ + \begin{split} + \sum_{i = 1}^{n} {x_i (Y_i - \hat{\beta_1} x_i)} + &= \sum_{i = 1}^{n} {x_i Y_i} - \sum_{i = 1}^{n} \hat{\beta_1} x_i^2 + \\ + &= \sum_{i = 1}^{n} {x_i Y_i} - \hat{\beta_1}\sum_{i = 1}^{n} x_i^2 + \end{split} + \] + + Solving for \(\hat{\beta_1}\) gives the final estimator for \(\beta_1\): + + \[ + \begin{split} + \hat{\beta_1} + &= \frac{ + \sum {x_i Y_i} + }{ + \sum x_i^2 + } + \end{split} + \] + + \pagebreak + + \part + + Calculate the bias and the variance for the estimated slope + \(\hat{\beta_1}\). + \\ + + \solution + + For the bias, we need to calculate the expected value + \(\E[\hat{\beta_1}]\): + + \[ + \begin{split} + \E[\hat{\beta_1}] + &= \E \left[ \frac{ + \sum {x_i Y_i} + }{ + \sum x_i^2 + }\right] + \\ + &= \frac{ + \sum {x_i \E[Y_i]} + }{ + \sum x_i^2 + } + \\ + &= \frac{ + \sum {x_i (\beta_1 x_i)} + }{ + \sum x_i^2 + } + \\ + &= \frac{ + \sum {x_i^2 \beta_1} + }{ + \sum x_i^2 + } + \\ + &= \beta_1 \frac{ + \sum {x_i^2 \beta_1} + }{ + \sum x_i^2 + } + \\ + &= \beta_1 + \end{split} + \] + + Thus since our estimator's expected value is \(\beta_1\), we can conclude + that the bias of our estimator is 0. + \\ + + For the variance: + + \[ + \begin{split} + \Var[\hat{\beta_1}] + &= \Var \left[ \frac{ + \sum {x_i Y_i} + }{ + \sum x_i^2 + }\right] + \\ + &= + \frac{ + \sum {x_i^2} + }{ + \sum x_i^2 \sum x_i^2 + } \Var[Y_i] + \\ + &= + \frac{ + \sum {x_i^2} + }{ + \sum x_i^2 \sum x_i^2 + } \Var[Y_i] + \\ + &= + \frac{ + 1 + }{ + \sum x_i^2 + } \Var[Y_i] + \\ + &= + \frac{ + 1 + }{ + \sum x_i^2 + } \sigma^2 + \\ + &= + \frac{ + \sigma^2 + }{ + \sum x_i^2 + } + \end{split} + \] + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Prove a polynomial of degree \(k\), \(a_kn^k + a_{k - 1}n^{k - 1} + \hdots + + a_1n^1 + a_0n^0\) is a member of \(\Theta(n^k)\) where \(a_k \hdots a_0\) + are nonnegative constants. + + \begin{proof} + To prove that \(a_kn^k + a_{k - 1}n^{k - 1} + \hdots + a_1n^1 + + a_0n^0\), we must show the following: + + \[ + \exists c_1 \exists c_2 \forall n \geq n_0,\ {c_1 \cdot g(n) \leq + f(n) \leq c_2 \cdot g(n)} + \] + + For the first inequality, it is easy to see that it holds because no + matter what the constants are, \(n^k \leq a_kn^k + a_{k - 1}n^{k - 1} + + \hdots + a_1n^1 + a_0n^0\) even if \(c_1 = 1\) and \(n_0 = 1\). This + is because \(n^k \leq c_1 \cdot a_kn^k\) for any nonnegative constant, + \(c_1\) and \(a_k\). + \\ + + Taking the second inequality, we prove it in the following way. + By summation, \(\sum\limits_{i=0}^k a_i\) will give us a new constant, + \(A\). By taking this value of \(A\), we can then do the following: + + \[ + \begin{split} + a_kn^k + a_{k - 1}n^{k - 1} + \hdots + a_1n^1 + a_0n^0 &= + \\ + &\leq (a_k + a_{k - 1} \hdots a_1 + a_0) \cdot n^k + \\ + &= A \cdot n^k + \\ + &\leq c_2 \cdot n^k + \end{split} + \] + + where \(n_0 = 1\) and \(c_2 = A\). \(c_2\) is just a constant. Thus the + proof is complete. + \end{proof} +\end{homeworkProblem} + +\pagebreak + +% +% Non sequential homework problems +% + +% Jump to problem 18 +\begin{homeworkProblem}[18] + Evaluate \(\sum_{k=1}^{5} k^2\) and \(\sum_{k=1}^{5} (k - 1)^2\). +\end{homeworkProblem} + +% Continue counting to 19 +\begin{homeworkProblem} + Find the derivative of \(f(x) = x^4 + 3x^2 - 2\) +\end{homeworkProblem} + +% Go back to where we left off +\begin{homeworkProblem}[6] + Evaluate the integrals + \(\int_0^1 (1 - x^2) \dx\) + and + \(\int_1^{\infty} \frac{1}{x^2} \dx\). +\end{homeworkProblem} +\end{comment} +\end{document}