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  65\pagestyle{fancy}
  66\fancyhead[LO,LE]{Year 12 Methods}
  67\fancyhead[CO,CE]{Andrew Lorimer}
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  91
  92\begin{document}
  93
  94\title{\vspace{-20mm}Year 12 Methods}
  95\author{Andrew Lorimer}
  96\date{}
  97\maketitle
  98
  99
 100\section{Functions}
 101
 102\begin{itemize} \tightlist
 103  \item vertical line test
 104  \item each \(x\) value produces only one \(y\) value
 105\end{itemize}
 106
 107\subsection*{One to one functions}
 108
 109\begin{itemize} \tightlist
 110  \item
 111    \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
 112    \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
 113    \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
 114    \(x^3\) is)
 115  \item
 116    horizontal line test
 117  \item
 118    if not one to one, it is many to one
 119\end{itemize}
 120
 121\subsection*{Odd and even functions}
 122
 123\begin{align*}
 124  \text{Even:}&& f(x)  &= f(-x) \\
 125  \text{Odd:} && -f(x) &= f(-x)
 126\end{align*}
 127
 128Even \(\implies\) symmetrical across \(y\)-axis \\
 129\(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
 130For \(x^n\), parity of \(n \equiv\) parity of function
 131
 132\begin{tabularx}{\columnwidth}{XX}
 133  \textbf{Even:} & \textbf{Odd:} \\
 134  \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^2)};  \end{axis}\end{tikzpicture} &
 135    \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^3)};  \end{axis}\end{tikzpicture}
 136\end{tabularx}
 137
 138\subsection*{Inverse functions}
 139
 140\begin{itemize} \tightlist
 141  \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\)
 142  \item \(f\) must be one to one
 143  \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
 144  \item Represents reflection across \(y=x\)
 145  \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\)
 146  \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\
 147    \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
 148  \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\
 149\end{itemize}
 150
 151\subsubsection*{Finding \(f^{-1}\)}
 152
 153\begin{enumerate} \tightlist
 154  \item Let \(y=f(x)\)
 155  \item Swap \(x\) and \(y\) (``take inverse''
 156  \item Solve for \(y\) \\
 157    Sqrt: state \(\pm\) solutions then restrict
 158  \item State rule as \(f^{-1}(x)=\dots\)
 159  \item For inverse \emph{function}, state in function notation
 160\end{enumerate}
 161
 162\subsection*{Simultaneous equations (linear)}
 163
 164\begin{itemize} \tightlist
 165  \item \textbf{Unique solution} - lines intersect at point
 166  \item \textbf{Infinitely many solutions} - lines are equal
 167  \item \textbf{No solution} - lines are parallel
 168\end{itemize}
 169
 170\subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions}
 171  where all coefficients are known except for one, and \(a, b\) are known
 172
 173  \begin{enumerate} \tightlist
 174    \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix}  \begin{bmatrix} x \\ y \end{bmatrix}  =  \begin{bmatrix} a \\ b \end{bmatrix}\)
 175      \item Find determinant of first matrix: \(\Delta = ps-qr\)
 176      \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
 177        or let \(\Delta \ne 0\) for one unique solution.
 178      \item Solve determinant equation to find variable \\
 179        \textbf{For infinite/no solutions:}
 180      \item Substitute variable into both original equations
 181      \item Rearrange equations so that LHS of each is the same
 182      \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\
 183        \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns)
 184  \end{enumerate}
 185
 186  \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
 187
 188  \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
 189
 190    \begin{itemize} \tightlist
 191      \item Use elimination
 192      \item Generate two new equations with only two variables
 193      \item Rearrange \& solve
 194      \item Substitute one variable into another equation to find another variable
 195    \end{itemize}
 196
 197    \subsection*{Piecewise functions}
 198
 199    \[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
 200
 201      \textbf{Open circle:} point included\\
 202      \textbf{Closed circle:} point not included
 203
 204      \subsection*{Operations on functions}
 205
 206      For \(f \pm g\) and \(f \times g\):
 207      \quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
 208
 209      Addition of linear piecewise graphs: add \(y\)-values at key points
 210
 211      Product functions:
 212
 213      \begin{itemize}
 214          \tightlist
 215        \item
 216          product will equal 0 if \(f=0\) or \(g=0\)
 217        \item
 218          \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
 219      \end{itemize}
 220
 221      \subsection*{Composite functions}
 222
 223      \((f \circ g)(x)\) is defined iff
 224      \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
 225
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 237        blankplot/.append style={orange, mark=none}
 238      }
 239
 240      \begin{figure*}[ht]
 241        \centering
 242
 243        \begin{tabularx}{\textwidth}{r|Y|Y}
 244
 245          & \(n\) is even & \(n\) is odd \\ \hline
 246
 247          \centering \(x^n, n \in \mathbb{Z}^+\) & 
 248
 249          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 250            \begin{axis}[blank, xmin=-3,  xmax=3]
 251              \addplot[blankplot] {(x^2)};
 252            \end{axis}
 253          \end{tikzpicture}} &
 254
 255          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 256            \begin{axis}[blank, xmin=-3,  xmax=3]
 257              \addplot[blankplot, domain=-3:3] {(x^3)};
 258            \end{axis}
 259          \end{tikzpicture}} \\ \hline
 260
 261          \centering \(x^n, n \in \mathbb{Z}^-\) &
 262
 263          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 264            \begin{axis}[blank, xmin=-4, xmax=4, ymax=8, ymin=-0]
 265              \addplot[blankplot, samples=100] {(x^(-2))};
 266            \end{axis}
 267          \end{tikzpicture}} &
 268
 269          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 270            \begin{axis}[blank, xmin=-3, xmax=3]
 271              \addplot[blankplot, domain=-3:-0.1] {(x^(-1))};
 272              \addplot[blankplot, domain=0.1:3] {(x^(-1))};
 273            \end{axis}
 274          \end{tikzpicture}} \\ \hline
 275
 276          \centering \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
 277
 278          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 279            \begin{axis}[blank, xmin=-1,  xmax=5]
 280              \addplot[blankplot] {(x^(1/2))};
 281            \end{axis}
 282          \end{tikzpicture}} &
 283
 284          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 285            \begin{axis}[blank, xmin=-3, xmax=3, ymin=-3, ymax=3]
 286              \addplot [blankplot, domain=-2:2] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
 287            \end{axis}
 288          \end{tikzpicture}} \\ \hline
 289
 290        \end{tabularx}
 291      \end{figure*}
 292
 293      \section{Polynomials}
 294
 295      \subsection*{Linear equations}
 296
 297      \subsubsection*{Forms}
 298
 299      \begin{itemize}
 300          \tightlist
 301        \item \(y=mx+c\)
 302        \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph
 303        \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts
 304      \end{itemize}
 305
 306      \subsubsection*{Line properties}
 307
 308      Parallel lines: \(m_1 = m_2\)\\
 309      Perpendicular lines: \(m_1 \times m_2 = -1\)\\
 310      Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
 311
 312      \subsection*{Quadratics}
 313      \setlength{\abovedisplayskip}{1pt}
 314      \setlength{\belowdisplayskip}{1pt}
 315      \[ x^2 + bx + c = (x+m)(x+n) \]
 316      \hfill where \(mn=c, \> m+n=b\)
 317
 318      \textbf{Difference of squares}
 319      \[ a^2 - b^2 = (a-b)(a+b) \]
 320      \textbf{Perfect squares}
 321      \[ a^2 \pm 2ab + b^2 = (a \pm b^2) \]
 322      \textbf{Completing the square}
 323      \begin{align*}
 324        x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\
 325        ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a}
 326      \end{align*}
 327      \textbf{Quadratic formula}
 328      \[ x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \]
 329      \hfill (Discriminant \(\Delta=b^2-4ac\))
 330
 331      \subsection*{Cubics}
 332
 333      \textbf{Difference of cubes}
 334      \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]
 335      \textbf{Sum of cubes}
 336      \[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]
 337      \textbf{Perfect cubes}
 338      \[ a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3 \]
 339
 340      \[ y=a(bx-h)^3 + c \]
 341
 342      \begin{itemize}
 343          \tightlist
 344        \item
 345          \(m=0\) at \emph{stationary point of inflection}
 346          (i.e.~(\({h \over b}, k)\))
 347        \item \(y=(x-a)^2(x-b)\) --- max at \(x=a\), min at \(x=b\)
 348        \item \(y=a(x-b)(x-c)(x-d)\) --- roots at \(b, c, d\)
 349        \item \(y=a(x-b)^2(x-c)\) --- roots at \(b\) (instantaneous), \(c\) (intercept)
 350      \end{itemize}
 351
 352      \subsection*{Quartic graphs}
 353
 354      \subsubsection*{Forms of quartic equations}
 355
 356      \(y=ax^4\)\\
 357      \(y=a(x-b)(x-c)(x-d)(x-e)\)\\
 358      \(y=ax^4+cd^2 (c \ge 0)\)\\
 359      \(y=ax^2(x-b)(x-c)\)\\
 360      \(y=a(x-b)^2(x-c)^2\)\\
 361      \(y=a(x-b)(x-c)^3\)
 362
 363      \input{transformations}
 364      \input{stuff}
 365      \input{circ-functions}
 366      \input{calculus}
 367
 368
 369
 370      \section{Statistics}
 371
 372      \subsection*{Probability}
 373
 374      \begin{align*}
 375        \Pr(A \cup B) &= \Pr(A) + \Pr(B) - \Pr(A \cap B) \\
 376        \Pr(A \cap B) &= \Pr(A|B) \times \Pr(B) \\
 377        \Pr(A|B) &= \frac{\Pr(A \cap B)}{\Pr(B)} \\
 378        \Pr(A) &= \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime})
 379      \end{align*}
 380
 381      Mutually exclusive \(\implies \Pr(A \cup B) = 0\) \\
 382
 383      Independent events:
 384      \begin{flalign*}
 385        \quad \Pr(A \cap B) &= \Pr(A) \times \Pr(B)& \\
 386        \Pr(A|B) &= \Pr(A) \\
 387        \Pr(B|A) &= \Pr(B)
 388      \end{flalign*}
 389
 390      \subsection*{Combinatorics}
 391
 392      \begin{itemize}
 393        \item Arrangements \({n \choose k} = \frac{n!}{(n-k)}\)
 394        \item \colorbox{important}{Combinations} \({n \choose k} = \frac{n!}{k!(n-k)!}\)
 395        \item Note \({n \choose k} = {n \choose k-1}\)
 396      \end{itemize}
 397
 398      \subsection*{Distributions}
 399
 400      \subsubsection*{Mean \(\mu\)}
 401
 402      \textbf{Mean} \(\mu\) or \textbf{expected value} \(E(X)\)
 403
 404      \begin{align*}
 405        E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f} \tag{\(f =\) absolute frequency} \\
 406        &= \sum_{i=1}^n \left[ x_i \cdot \Pr(X=x_i) \right] \tag{discrete}\\
 407        &= \int_\textbf{X} (x \cdot f(x)) \> dx
 408      \end{align*}
 409
 410      \subsubsection*{Mode}
 411
 412      Most popular value (has highest probability of all \(X\) values). Multiple modes can exist if \(>1 \> X\) value have equal-highest probability. Number must exist in distribution.
 413
 414      \subsubsection*{Median}
 415
 416      If \(m > 0.5\), then value of \(X\) that is reached is the median of \(X\). If \(m = 0.5 = 0.5\), then \(m\) is halfway between this value and the next. To find \(m\), add values of \(X\) from smallest to alrgest until the sum reaches 0.5.
 417
 418      \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \]
 419
 420      \subsubsection*{Variance \(\sigma^2\)}
 421
 422      \begin{align*}
 423        \operatorname{Var}(x) &= \sum_{i=1}^n p_i (x_i-\mu)^2 \\
 424        &= \sum (x-\mu)^2 \times \Pr(X=x) \\
 425        &= \sum x^2 \times p(x) - \mu^2 \\
 426        &= \operatorname{E}(X^2) - [\operatorname{E}(X)]^2
 427        &= E\left[(X-\mu)^2\right]
 428      \end{align*}
 429
 430      \subsubsection*{Standard deviation \(\sigma\)}
 431
 432      \begin{align*}
 433        \sigma &= \operatorname{sd}(X) \\
 434        &= \sqrt{\operatorname{Var}(X)}
 435      \end{align*}
 436
 437      \subsection*{Binomial distributions}
 438
 439      Conditions for a \textit{binomial distribution}:
 440      \begin{enumerate}
 441        \item Two possible outcomes: \textbf{success} or \textbf{failure}
 442        \item \(\Pr(\text{success})\) is constant across trials (also denoted \(p\))
 443        \item Finite number \(n\) of independent trials
 444      \end{enumerate}
 445
 446
 447      \subsubsection*{Properties of \(X \sim \operatorname{Bi}(n,p)\)}
 448
 449      \begin{align*}
 450        \mu(X) &= np \\
 451        \operatorname{Var}(X) &= np(1-p) \\
 452        \sigma(X) &= \sqrt{np(1-p)} \\
 453        \Pr(X=x) &= {n \choose x} \cdot p^x \cdot (1-p)^{n-x}
 454      \end{align*}
 455
 456      \begin{cas}
 457        Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPdf; then input
 458        \begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=3cm, font=\normalfont]
 459          \item [x:] no. of successes
 460          \item [numtrial:] no. of trials
 461          \item [pos:] probability of success
 462        \end{description}
 463      \end{cas}
 464
 465      \subsection*{Continuous random variables}
 466
 467      A continuous random variable \(X\) has a pdf \(f\) such that:
 468
 469      \begin{enumerate}
 470        \item \(f(x) \ge 0 \forall x \)
 471        \item \(\int^\infty_{-\infty} f(x) \> dx = 1\)
 472      \end{enumerate}
 473
 474      \begin{align*}
 475        E(X) &= \int_\textbf{X} (x \cdot f(x)) \> dx \\
 476        \operatorname{Var}(X) &= E\left[(X-\mu)^2\right]
 477      \end{align*}
 478
 479      \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]
 480
 481
 482      \subsection*{Two random variables \(X, Y\)}
 483
 484      If \(X\) and \(Y\) are independent:
 485      \begin{align*}
 486        \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\
 487        \operatorname{Var}(aX \pm bY \pm c) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y)
 488      \end{align*}
 489
 490      \subsection*{Linear functions \(X \rightarrow aX+b\)}
 491
 492      \begin{align*}
 493        \Pr(Y \le y) &= \Pr(aX+b \le y) \\
 494        &= \Pr\left(X \le \dfrac{y-b}{a}\right) \\
 495        &= \int^{\frac{y-b}{a}}_{-\infty} f(x) \> dx
 496      \end{align*}
 497
 498      \begin{align*}
 499        \textbf{Mean:} && \operatorname{E}(aX+b) & = a\operatorname{E}(X)+b \\
 500        \textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2 \operatorname{Var}(X) \\
 501      \end{align*}
 502
 503      \subsection*{Expectation theorems}
 504
 505      For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\).
 506
 507      \begin{align*}
 508        E(X^2) &= \operatorname{Var}(X) - \left[E(X)\right]^2 \\
 509        E(X^n) &= \Sigma x^n \cdot p(x) \tag{non-linear} \\
 510        &\ne [E(X)]^n \\
 511        E(aX \pm b) &= aE(X) \pm b \tag{linear} \\
 512        E(b) &= b \tag{\(\forall b \in \mathbb{R}\)}\\
 513        E(X+Y) &= E(X) + E(Y) \tag{two variables}
 514      \end{align*}
 515
 516      \subsection*{Sample mean}
 517
 518      Approximation of the \textbf{population mean} determined experimentally.
 519
 520      \[ \overline{x} = \dfrac{\Sigma x}{n} \]
 521
 522      where
 523      \begin{description}[nosep, labelindent=0.5cm]
 524        \item \(n\) is the size of the sample (number of sample points)
 525        \item \(x\) is the value of a sample point
 526      \end{description}
 527
 528      \begin{cas}
 529        \begin{enumerate}[leftmargin=3mm]
 530          \item Spreadsheet
 531          \item In cell A1:\\ \path{mean(randNorm(sd, mean, sample size))}
 532          \item Edit \(\rightarrow\) Fill \(\rightarrow\) Fill Range
 533          \item Input range as A1:An where \(n\) is the number of samples
 534          \item Graph \(\rightarrow\) Histogram
 535        \end{enumerate}
 536      \end{cas}
 537
 538      \subsubsection*{Sample size of \(n\)}
 539
 540      \[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \]
 541
 542      Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\) (approaches these values for increasing sample size \(n\)).
 543
 544      For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \operatorname{E}(X), \quad \operatorname{sd}(X^\prime) = \dfrac{\operatorname{sd}(X)}{\sqrt{n}}\)
 545
 546      \begin{cas}
 547
 548        \begin{itemize}
 549          \item Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); where \verb;n; is the number of samples. Show histogram with Histogram key in top left
 550          \item To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable
 551        \end{itemize}
 552
 553      \end{cas}
 554
 555      \subsection*{Normal distributions}
 556
 557
 558      \[ Z = \frac{X - \mu}{\sigma} \]
 559
 560      Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\) \\
 561      \(\text{mean} = \text{mode} = \text{median}\)
 562
 563      \begin{warning}
 564        Always express \(z\) as +ve. Express confidence \textit{interval} as ordered pair.
 565      \end{warning}
 566
 567      \pgfmathdeclarefunction{gauss}{2}{%
 568        \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
 569        }
 570        \pgfkeys{/pgf/decoration/.cd,
 571        distance/.initial=10pt
 572        }  \pgfdeclaredecoration{add dim}{final}{
 573          \state{final}{% 
 574            \pgfmathsetmacro{\dist}{5pt*\pgfkeysvalueof{/pgf/decoration/distance}/abs(\pgfkeysvalueof{/pgf/decoration/distance})}    
 575            \pgfpathmoveto{\pgfpoint{0pt}{0pt}}             
 576            \pgfpathlineto{\pgfpoint{0pt}{2*\dist}}   
 577            \pgfpathmoveto{\pgfpoint{\pgfdecoratedpathlength}{0pt}} 
 578            \pgfpathlineto{\pgfpoint{(\pgfdecoratedpathlength}{2*\dist}}     
 579            \pgfsetarrowsstart{latex}
 580            \pgfsetarrowsend{latex}
 581            \pgfpathmoveto{\pgfpoint{0pt}{\dist}}
 582            \pgfpathlineto{\pgfpoint{\pgfdecoratedpathlength}{\dist}} 
 583            \pgfusepath{stroke} 
 584            \pgfpathmoveto{\pgfpoint{0pt}{0pt}}
 585            \pgfpathlineto{\pgfpoint{\pgfdecoratedpathlength}{0pt}}
 586            }}
 587            \tikzset{dim/.style args={#1,#2}{decoration={add dim,distance=#2},
 588            decorate,
 589            postaction={decorate,decoration={text along path,
 590            raise=#2,
 591            text align={align=center},
 592            text={#1}}}}}
 593            \begin{figure*}[hb]
 594              \centering
 595              \begin{tikzpicture}
 596                \begin{axis}[every axis plot post/.style={
 597                    mark=none,domain=-3:3,samples=50,smooth}, 
 598                  axis x line=bottom, 
 599                  axis y line=left,
 600                  enlargelimits=upper,
 601                  x=\textwidth/10,
 602                  ytick={0.55},
 603                  yticklabels={\(\frac{1}{\sigma \sqrt{2\pi}}\)}, 
 604                  xtick={-2,-1,0,1,2},
 605                  x tick label style = {font=\footnotesize},
 606                  xticklabels={\((\mu-2\sigma)\), \((\mu-\sigma)\), \(\mu\), \((\mu+\sigma)\), \((\mu+2\sigma)\)},
 607                  xlabel={\(x\)},
 608                  every axis x label/.style={at={(current axis.right of origin)},anchor=north west},
 609                  every axis y label/.style={at={(axis description cs:-0.02,0.2)}, anchor=south west, rotate=90},
 610                  ylabel={\(\Pr(X=x)\)}]
 611                  \addplot {gauss(0,0.75)};
 612                  \fill[red!30] (-3,0)  -- plot[id=f3,domain=-3:3,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-((x)^2)/(2*0.75^2))} -- (3,0) -- cycle;
 613                  \fill[darkgray!30] (3,0)  -- plot[id=f3,domain=-3:3,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (3,0) -- cycle;
 614                  \fill[lightgray!30] (-2,0)  -- plot[id=f3,domain=-2:2,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (2,0) -- cycle;
 615                  \fill[white!30] (-1,0)  -- plot[id=f3,domain=-1:1,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (1,0) -- cycle;
 616                  \begin{scope}[<->]
 617                    \draw (-1,0.35) -- (1,0.35) node [midway, fill=white] {68.3\%};
 618                    \draw (-2,0.25) -- (2,0.25) node [midway, fill=white] {95.5\%};
 619                    \draw (-3,0.15) -- (3,0.15) node [midway, fill=white] {99.7\%};
 620                  \end{scope}
 621                  \begin{scope}[-, dashed, gray]
 622                    \draw (-1,0) -- (-1, 0.35);
 623                    \draw (1,0) -- (1, 0.35);
 624                    \draw (-2,0) -- (-2, 0.25);
 625                    \draw (2,0) -- (2, 0.25);
 626                    \draw (-3,0) -- (-3, 0.15);
 627                    \draw (3,0) -- (3, 0.15);
 628                  \end{scope}
 629                \end{axis}
 630                \begin{axis}[every axis plot post/.append style={
 631                    mark=none,domain=-3:3,samples=50,smooth}, 
 632                  axis x line=bottom, 
 633                  enlargelimits=upper,
 634                  x=\textwidth/10,
 635                  xtick={-2,-1,0,1,2},
 636                  axis x line shift=30pt,
 637                  hide y axis,
 638                  x tick label style = {font=\footnotesize},
 639                  xlabel={\(Z\)},
 640                  every axis x label/.style={at={(axis description cs:1,-0.25)},anchor=south west}]
 641                  \addplot {gauss(0,0.75)};
 642                \end{axis}
 643              \end{tikzpicture}
 644            \end{figure*}
 645          \end{document}