From 4f670b750af5c11e1eac16d9cd8556455f89f46a Mon Sep 17 00:00:00 2001
From: Juan Marín Noguera <juan@mnpi.eu>
Date: Fri, 16 May 2025 22:18:44 +0200
Subject: Changed layout for more manageable volumes

---
 3.4.1.lyx | 1677 -------------------------------------------------------------
 1 file changed, 1677 deletions(-)
 delete mode 100644 3.4.1.lyx

(limited to '3.4.1.lyx')

diff --git a/3.4.1.lyx b/3.4.1.lyx
deleted file mode 100644
index 2cd7261..0000000
--- a/3.4.1.lyx
+++ /dev/null
@@ -1,1677 +0,0 @@
-#LyX 2.4 created this file. For more info see https://www.lyx.org/
-\lyxformat 620
-\begin_document
-\begin_header
-\save_transient_properties true
-\origin unavailable
-\textclass book
-\begin_preamble
-\input defs
-\end_preamble
-\use_default_options true
-\maintain_unincluded_children no
-\language english
-\language_package default
-\inputencoding utf8
-\fontencoding auto
-\font_roman "default" "default"
-\font_sans "default" "default"
-\font_typewriter "default" "default"
-\font_math "auto" "auto"
-\font_default_family default
-\use_non_tex_fonts false
-\font_sc false
-\font_roman_osf false
-\font_sans_osf false
-\font_typewriter_osf false
-\font_sf_scale 100 100
-\font_tt_scale 100 100
-\use_microtype false
-\use_dash_ligatures true
-\graphics default
-\default_output_format default
-\output_sync 0
-\bibtex_command default
-\index_command default
-\float_placement class
-\float_alignment class
-\paperfontsize default
-\spacing single
-\use_hyperref false
-\papersize default
-\use_geometry false
-\use_package amsmath 1
-\use_package amssymb 1
-\use_package cancel 1
-\use_package esint 1
-\use_package mathdots 1
-\use_package mathtools 1
-\use_package mhchem 1
-\use_package stackrel 1
-\use_package stmaryrd 1
-\use_package undertilde 1
-\cite_engine basic
-\cite_engine_type default
-\biblio_style plain
-\use_bibtopic false
-\use_indices false
-\paperorientation portrait
-\suppress_date false
-\justification true
-\use_refstyle 1
-\use_formatted_ref 0
-\use_minted 0
-\use_lineno 0
-\index Index
-\shortcut idx
-\color #008000
-\end_index
-\secnumdepth 3
-\tocdepth 3
-\paragraph_separation indent
-\paragraph_indentation default
-\is_math_indent 0
-\math_numbering_side default
-\quotes_style english
-\dynamic_quotes 0
-\papercolumns 1
-\papersides 1
-\paperpagestyle default
-\tablestyle default
-\tracking_changes false
-\output_changes false
-\change_bars false
-\postpone_fragile_content false
-\html_math_output 0
-\html_css_as_file 0
-\html_be_strict false
-\docbook_table_output 0
-\docbook_mathml_prefix 1
-\end_header
-
-\begin_body
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-exerc1[10]
-\end_layout
-
-\end_inset
-
-If 
-\begin_inset Formula $\alpha$
-\end_inset
-
- and 
-\begin_inset Formula $\beta$
-\end_inset
-
- are real numbers with 
-\begin_inset Formula $\alpha<\beta$
-\end_inset
-
-,
- how would you generate a random real number uniformly distributed between 
-\begin_inset Formula $\alpha$
-\end_inset
-
- and 
-\begin_inset Formula $\beta$
-\end_inset
-
-?
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-By taking a uniformly distributed real number 
-\begin_inset Formula $U$
-\end_inset
-
- between 0 and 1 and returning 
-\begin_inset Formula $\alpha+(\beta-\alpha)U$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-rexerc3[14]
-\end_layout
-
-\end_inset
-
-Discuss treating 
-\begin_inset Formula $U$
-\end_inset
-
- as an integer and computing its 
-\emph on
-remainder
-\emph default
- 
-\begin_inset Formula $\bmod k$
-\end_inset
-
- to get a random integer between 0 and 
-\begin_inset Formula $k-1$
-\end_inset
-
-,
- instead of multiplying as suggested in the text.
- Thus (1) would be changed to
-\begin_inset Formula 
-\begin{align*}
- & \mathtt{ENTA\ 0}; &  & \mathtt{LDX\ U}; &  & \mathtt{DIV\ K},
-\end{align*}
-
-\end_inset
-
-with the result appearing in register X.
- Is this a good method?
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-While in theory it's as good as the other one,
- less significant bits tend to be less random in many random number generation methods,
- so we prefer the method in the text.
- This is specially important if we use a linear congruential method where 
-\begin_inset Formula $k$
-\end_inset
-
- is not relatively prime to 
-\begin_inset Formula $m$
-\end_inset
-
-.
- In addition,
- if 
-\begin_inset Formula $k$
-\end_inset
-
- is not much smaller than 
-\begin_inset Formula $m$
-\end_inset
-
-,
- numbers lower than 
-\begin_inset Formula $m\bmod k$
-\end_inset
-
- will have a larger probability of appearing,
- whereas in the method of the text a similar effect is observed but the numbers with higher probability are more evenly distributed.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-rexerc5[21]
-\end_layout
-
-\end_inset
-
-Suggest an efficient way to compute a random variable with the distribution 
-\begin_inset Formula $F(x)=px+qx^{2}+rx^{3}$
-\end_inset
-
-,
- where 
-\begin_inset Formula $p\geq0$
-\end_inset
-
-,
- 
-\begin_inset Formula $q\geq0$
-\end_inset
-
-,
- 
-\begin_inset Formula $r\geq0$
-\end_inset
-
-,
- and 
-\begin_inset Formula $p+q+r=1$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-Assume this formula for the distribution is valid for 
-\begin_inset Formula $x\in[0,1]$
-\end_inset
-
-,
- as 
-\begin_inset Formula $F(0)=0$
-\end_inset
-
- and 
-\begin_inset Formula $F(1)=1$
-\end_inset
-
-.
- Now,
- 
-\begin_inset Formula $F(x)=x(rx^{2}+qx+p)$
-\end_inset
-
-,
- or more precisely,
-\begin_inset Formula 
-\[
-F(x)=\max\{0,\min\{1,x\}\}\cdot\begin{cases}
-0, & x\leq-\tfrac{q}{2r};\\
-\min\{1,rx^{2}+qx+p\}, & x\geq-\tfrac{q}{2r},
-\end{cases}
-\]
-
-\end_inset
-
-so we may take 
-\begin_inset Formula $U_{1}$
-\end_inset
-
- and 
-\begin_inset Formula $U_{2}$
-\end_inset
-
- uniformly distributed between 0 and 1 and return
-\begin_inset Formula 
-\[
-\max\left\{ U_{1},-\frac{q}{2r}+\sqrt{\left(\frac{q}{2r}\right)^{2}-\frac{p}{r}+\frac{U_{2}}{r}}\right\} .
-\]
-
-\end_inset
-
-Note that we take the 
-\begin_inset Quotes eld
-\end_inset
-
-
-\begin_inset Formula $+$
-\end_inset
-
-
-\begin_inset Quotes erd
-\end_inset
-
- sign in the solution to the quadratic equation to choose the solution in the right side of the parabola.
- The method in the book is more efficient.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-rexerc7[20]
-\end_layout
-
-\end_inset
-
-(A.
- J.
- Walker) Suppose we have a bunch of cubes of 
-\begin_inset Formula $k$
-\end_inset
-
- different colors,
- say 
-\begin_inset Formula $n_{j}$
-\end_inset
-
- cubes of color 
-\begin_inset Formula $C_{j}$
-\end_inset
-
- for 
-\begin_inset Formula $1\leq j\leq k$
-\end_inset
-
-,
- and we also have 
-\begin_inset Formula $k$
-\end_inset
-
- boxes 
-\begin_inset Formula $\{B_{1},\dots,B_{k}\}$
-\end_inset
-
- each of which can hold exactly 
-\begin_inset Formula $n$
-\end_inset
-
- cubes.
- Furthermore 
-\begin_inset Formula $n_{1}+\dots+n_{k}=kn$
-\end_inset
-
-,
- so the cubes will just fit into the boxes.
- Prove (constructively) that there is always a way to put the cubes into the boxes so that each box contains at most two different colors of cubes;
- in fact,
- there is a way to do it so that,
- wherever box 
-\begin_inset Formula $B_{j}$
-\end_inset
-
- contains two colors,
- one of those colors is 
-\begin_inset Formula $C_{j}$
-\end_inset
-
-.
- Show how to use this principle to compute the 
-\begin_inset Formula $P$
-\end_inset
-
- and 
-\begin_inset Formula $Y$
-\end_inset
-
- tables required in (3),
- given a probability distribution 
-\begin_inset Formula $(p_{1},\dots,p_{k})$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-We prove this by induction on 
-\begin_inset Formula $k$
-\end_inset
-
-.
- For 
-\begin_inset Formula $k=1$
-\end_inset
-
- this is trivial.
- For 
-\begin_inset Formula $k>1$
-\end_inset
-
-,
- if there is a 
-\begin_inset Formula $j$
-\end_inset
-
- such that 
-\begin_inset Formula $n=n_{j}$
-\end_inset
-
-,
- we pack the 
-\begin_inset Formula $n_{j}$
-\end_inset
-
- cubes of color 
-\begin_inset Formula $C_{j}$
-\end_inset
-
- into 
-\begin_inset Formula $B_{j}$
-\end_inset
-
-;
- otherwise there exists 
-\begin_inset Formula $i$
-\end_inset
-
- and 
-\begin_inset Formula $j$
-\end_inset
-
- such that 
-\begin_inset Formula $n_{i}<n<n_{j}$
-\end_inset
-
-,
- so we pack the 
-\begin_inset Formula $n_{i}$
-\end_inset
-
- cubes of color 
-\begin_inset Formula $C_{i}$
-\end_inset
-
- and also 
-\begin_inset Formula $n-n_{i}$
-\end_inset
-
- cubes of color 
-\begin_inset Formula $C_{j}$
-\end_inset
-
- into 
-\begin_inset Formula $B_{i}$
-\end_inset
-
-.
- Either way we are left with 
-\begin_inset Formula $k-1$
-\end_inset
-
- colors and 
-\begin_inset Formula $k-1$
-\end_inset
-
- boxes.
-\end_layout
-
-\begin_layout Standard
-To construct the tables,
- we note that this proof doesn't require 
-\begin_inset Formula $n$
-\end_inset
-
- and the 
-\begin_inset Formula $n_{j}$
-\end_inset
-
-s to be integers,
- they can be arbitrary nonnegative nonnegative reals as long as the conditions are met,
- so we can let 
-\begin_inset Formula $n=\frac{1}{k}$
-\end_inset
-
- and 
-\begin_inset Formula $n_{j}=p_{j}$
-\end_inset
-
- (the probability of event 
-\begin_inset Formula $x_{j}$
-\end_inset
-
-) for each 
-\begin_inset Formula $j$
-\end_inset
-
-.
- Then,
- if after proceeding as in this proof,
- we find that 
-\begin_inset Formula $B_{i}$
-\end_inset
-
- has 
-\begin_inset Formula $p$
-\end_inset
-
- cubes of color 
-\begin_inset Formula $C_{i}$
-\end_inset
-
- and 
-\begin_inset Formula $n-p$
-\end_inset
-
- cubes of some other color 
-\begin_inset Formula $C_{j}$
-\end_inset
-
- (where 
-\begin_inset Formula $p\in[0,n]$
-\end_inset
-
-),
- we set 
-\begin_inset Formula $P=pk$
-\end_inset
-
- and 
-\begin_inset Formula $Y=j$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-rexerc10[HM24]
-\end_layout
-
-\end_inset
-
-Explain how to calculate auxiliary constants 
-\begin_inset Formula $P_{j}$
-\end_inset
-
-,
- 
-\begin_inset Formula $Q_{j}$
-\end_inset
-
-,
- 
-\begin_inset Formula $Y_{j}$
-\end_inset
-
-,
- 
-\begin_inset Formula $Z_{j}$
-\end_inset
-
-,
- 
-\begin_inset Formula $S_{j}$
-\end_inset
-
-,
- 
-\begin_inset Formula $D_{j}$
-\end_inset
-
-,
- 
-\begin_inset Formula $E_{j}$
-\end_inset
-
- so that Algorithm M delivers answers with the correct distribution.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-Let 
-\begin_inset Formula $p_{1},\dots,p_{31}$
-\end_inset
-
- be the probabilities as in the text and 
-\begin_inset Formula $p_{0}=0$
-\end_inset
-
-.
- We may calculate probabilities 
-\begin_inset Formula $p_{16},\dots,p_{30}$
-\end_inset
-
- by a numeral method like Simpson's rule,
- and this gives us values that are all lower than 
-\begin_inset Formula $\frac{1}{32}$
-\end_inset
-
-.
- 
-\end_layout
-
-\begin_layout Standard
-First,
- calculate 
-\begin_inset Formula $p_{1},\dots,p_{31}$
-\end_inset
-
- as defined in the text,
- obtaining that 
-\begin_inset Formula $p_{16},\dots,p_{31}<\frac{1}{32}$
-\end_inset
-
- (we may use some numerical method like the Simpson's rule),
- and set 
-\begin_inset Formula $p_{0}=0$
-\end_inset
-
-.
- Then,
- operate like in Exercise 7 for these probabilities to obtain tables 
-\begin_inset Formula $(P_{j},Y'_{j})_{j=0}^{31}$
-\end_inset
-
-,
- ensuring that 
-\begin_inset Formula $p_{15},\dots,p_{31}$
-\end_inset
-
- are considered first so that all the 
-\begin_inset Formula $Y'_{j}\in\{1,\dots,15\}$
-\end_inset
-
-.
- This gives us the 
-\begin_inset Formula $P_{j}$
-\end_inset
-
-,
- and then we set 
-\begin_inset Formula $Y_{j}=\frac{Y'_{j}-1}{5}$
-\end_inset
-
- and 
-\begin_inset Formula $Z_{j}=\frac{1}{5(1-P_{j})}$
-\end_inset
-
- for 
-\begin_inset Formula $0\leq j\leq31$
-\end_inset
-
-.
- We also set 
-\begin_inset Formula $S_{j}=\frac{j-1}{5}$
-\end_inset
-
- for 
-\begin_inset Formula $1\leq j\leq16$
-\end_inset
-
- and 
-\begin_inset Formula $Q_{j}=\frac{1}{5P_{j}}$
-\end_inset
-
- for 
-\begin_inset Formula $1\leq j\leq15$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Now,
- the wedge functions are
-\begin_inset Formula 
-\[
-f_{j+15}(x)\coloneqq\sqrt{\frac{2}{\pi}}\left(\text{e}^{-x^{2}/2}-\text{e}^{-j^{2}/50}\right)\bigg/p_{j+15}
-\]
-
-\end_inset
-
-for 
-\begin_inset Formula $1\leq j\leq15$
-\end_inset
-
-.
- For 
-\begin_inset Formula $1\leq j\leq5$
-\end_inset
-
-,
- the curve is concave,
- so we set
-\begin_inset Formula 
-\begin{align*}
-b_{j} & =-f'_{j+15}(S_{j+1})=\frac{j}{5p_{j+15}}\sqrt{\frac{2}{\pi}}\text{e}^{-j^{2}/50}, & a_{j} & =f_{j+15}(S_{j}).
-\end{align*}
-
-\end_inset
-
-For 
-\begin_inset Formula $6\leq j\leq15$
-\end_inset
-
- the curve is convex and we set 
-\begin_inset Formula $b_{j}=5f_{j+15}(S_{j})$
-\end_inset
-
- (the slope) and 
-\begin_inset Formula $a_{j}=f_{j+15}(x)+(x-S_{j})b$
-\end_inset
-
-,
- where 
-\begin_inset Formula $x\in[S_{j},S_{j+1}]$
-\end_inset
-
- is such that 
-\begin_inset Formula $f'_{j+15}(x)=-b_{j}$
-\end_inset
-
-.
- Now,
-\begin_inset Formula 
-\begin{multline*}
-f'_{j+15}(x)=-\sqrt{\frac{2}{\pi}}\frac{x\text{e}^{-x^{2}/2}}{p_{j+15}}=-\frac{5}{p_{j+15}}\sqrt{\frac{2}{\pi}}(\text{e}^{-(j-1)^{2}/50}-\text{e}^{-j^{2}/50})=-b_{j}\iff\\
-x\text{e}^{-x^{2}/2}=5(\text{e}^{-(j-1)^{2}/50}-\text{e}^{-j^{2}/50}).
-\end{multline*}
-
-\end_inset
-
-We know this value exists by Lagrange's mean value theorem,
- although we need to calculate it numerically (for example,
- by Newton's method,
- using that 
-\begin_inset Formula $\frac{\text{d}}{\text{d}x}(x\text{e}^{-x^{2}/2})=(1-x^{2})\text{e}^{-x^{2}/2}$
-\end_inset
-
-).
- Then,
- for for 
-\begin_inset Formula $1\leq j\leq15$
-\end_inset
-
-,
- we set 
-\begin_inset Formula $D_{j+15}=a_{j}/b_{j}$
-\end_inset
-
- and 
-\begin_inset Formula $E_{j+15}=\sqrt{\frac{2}{\pi}}\frac{e^{-j^{2}/50}}{b_{j}p_{j+15}}$
-\end_inset
-
-,
- so
-\begin_inset Formula 
-\[
-E_{j+15}=\begin{cases}
-\frac{5}{j}, & 1\leq j\leq5;\\
-\frac{5}{\text{e}^{(2j-1)/50}-1}, & 6\leq j\leq15.
-\end{cases}
-\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-\begin_inset Note Comment
-status open
-
-\begin_layout Plain Layout
-\begin_inset listings
-inline false
-status open
-
-\begin_layout Plain Layout
-
-use POSIX;
-\end_layout
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\begin_layout Plain Layout
-
-my (@D,
- @E,
- @P,
- @Q,
- @S,
- @Y,
- @Z);
-\end_layout
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\begin_layout Plain Layout
-
-sub normal {
-\end_layout
-
-\begin_layout Plain Layout
-
-    my $rand  = rand;
-\end_layout
-
-\begin_layout Plain Layout
-
-    my $sign  = POSIX::floor( $rand * 2 );
-\end_layout
-
-\begin_layout Plain Layout
-
-    my $piece = POSIX::floor( $rand * 64 ) % 32;
-\end_layout
-
-\begin_layout Plain Layout
-
-    my $dev   = ( $rand * 64 ) % 1;
-\end_layout
-
-\begin_layout Plain Layout
-
-    my $abs;
-\end_layout
-
-\begin_layout Plain Layout
-
-    if ( $dev >= $P[$piece] ) {
-\end_layout
-
-\begin_layout Plain Layout
-
-        $abs = $Y[$piece] + $dev * $Z[$piece];
-\end_layout
-
-\begin_layout Plain Layout
-
-    }
-\end_layout
-
-\begin_layout Plain Layout
-
-    elsif ( $piece <= 15 ) {
-\end_layout
-
-\begin_layout Plain Layout
-
-        $abs = $S[$piece] + $dev * $Q[$piece];
-\end_layout
-
-\begin_layout Plain Layout
-
-    }
-\end_layout
-
-\begin_layout Plain Layout
-
-    elsif ( $piece < 31 ) {
-\end_layout
-
-\begin_layout Plain Layout
-
-        $piece -= 16;
-\end_layout
-
-\begin_layout Plain Layout
-
-        my ( $u,
- $v );
-\end_layout
-
-\begin_layout Plain Layout
-
-        do {
-\end_layout
-
-\begin_layout Plain Layout
-
-            ( $u,
- $v ) = ( rand,
- rand );
-\end_layout
-
-\begin_layout Plain Layout
-
-            ( $u,
- $v ) = ( $v,
- $u ) if $u > $v;
-\end_layout
-
-\begin_layout Plain Layout
-
-            $abs = $S[$piece] + $u / 5;
-\end_layout
-
-\begin_layout Plain Layout
-
-        } while ( $v < $D[$piece]
-\end_layout
-
-\begin_layout Plain Layout
-
-            || $v <= $u +
-\end_layout
-
-\begin_layout Plain Layout
-
-            $E[$piece] * ( exp( ( $S[ $piece + 1 ]**2 - $abs**2 ) / 2 ) - 1 ) );
-\end_layout
-
-\begin_layout Plain Layout
-
-    }
-\end_layout
-
-\begin_layout Plain Layout
-
-    else {
-\end_layout
-
-\begin_layout Plain Layout
-
-        do { $abs = sqrt( 9 - 2 * log rand ) } while ( $abs * rand ) >= 3;
-\end_layout
-
-\begin_layout Plain Layout
-
-    }
-\end_layout
-
-\begin_layout Plain Layout
-
-    return $sign ?
- -$abs :
- $abs;
-\end_layout
-
-\begin_layout Plain Layout
-
-}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset listings
-inline false
-status open
-
-\begin_layout Plain Layout
-
-local D = ...
-\end_layout
-
-\begin_layout Plain Layout
-
-local E = ...
-\end_layout
-
-\begin_layout Plain Layout
-
-local P = ...
-\end_layout
-
-\begin_layout Plain Layout
-
-local Q = ...
-\end_layout
-
-\begin_layout Plain Layout
-
-local S = ...
-\end_layout
-
-\begin_layout Plain Layout
-
-local Y = ...
-\end_layout
-
-\begin_layout Plain Layout
-
-local Z = ...
-\end_layout
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\begin_layout Plain Layout
-
-local function normal ()
-\end_layout
-
-\begin_layout Plain Layout
-
-    local rand = math.random()
-\end_layout
-
-\begin_layout Plain Layout
-
-    local sign = math.floor(rand * 2)
-\end_layout
-
-\begin_layout Plain Layout
-
-    local piece = math.floor(rand * 64) % 32
-\end_layout
-
-\begin_layout Plain Layout
-
-    local dev = (rand * 64) % 1
-\end_layout
-
-\begin_layout Plain Layout
-
-    local abs
-\end_layout
-
-\begin_layout Plain Layout
-
-    if dev >= P[piece] then
-\end_layout
-
-\begin_layout Plain Layout
-
-        abs = Y[piece] + dev * Z[piece]
-\end_layout
-
-\begin_layout Plain Layout
-
-    elseif piece <= 15 then
-\end_layout
-
-\begin_layout Plain Layout
-
-        abs = S[piece] + dev * Q[piece]
-\end_layout
-
-\begin_layout Plain Layout
-
-    elseif piece < 31 then
-\end_layout
-
-\begin_layout Plain Layout
-
-        local u,
- v
-\end_layout
-
-\begin_layout Plain Layout
-
-        repeat
-\end_layout
-
-\begin_layout Plain Layout
-
-            u,
- v = math.random(),
- math.random()
-\end_layout
-
-\begin_layout Plain Layout
-
-            if u > v then u,
- v = v,
- u end
-\end_layout
-
-\begin_layout Plain Layout
-
-            abs = S[piece-15] + u/5
-\end_layout
-
-\begin_layout Plain Layout
-
-        until v >= D[piece-15] or
-\end_layout
-
-\begin_layout Plain Layout
-
-            v > u + E[piece-15] * (math.exp((S[piece-14]^2 - abs^2) / 2) - 1)
-\end_layout
-
-\begin_layout Plain Layout
-
-    else
-\end_layout
-
-\begin_layout Plain Layout
-
-        repeat
-\end_layout
-
-\begin_layout Plain Layout
-
-            abs = math.sqrt(9 - 2 * math.log(math.random()))
-\end_layout
-
-\begin_layout Plain Layout
-
-        until abs * rand >= 3
-\end_layout
-
-\begin_layout Plain Layout
-
-    end
-\end_layout
-
-\begin_layout Plain Layout
-
-    if sign == 0 then return abs else return -abs end
-\end_layout
-
-\begin_layout Plain Layout
-
-end
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset listings
-inline false
-status open
-
-\begin_layout Plain Layout
-
-from math import exp,
- floor,
- log,
- sqrt
-\end_layout
-
-\begin_layout Plain Layout
-
-from random import random
-\end_layout
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\begin_layout Plain Layout
-
-D,
- E,
- P,
- Q,
- S,
- Y,
- Z = [],
- [],
- [],
- [],
- [],
- [],
- []
-\end_layout
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\begin_layout Plain Layout
-
-def normal():
-\end_layout
-
-\begin_layout Plain Layout
-
-    rand = random()
-\end_layout
-
-\begin_layout Plain Layout
-
-    sign = floor(rand * 2)
-\end_layout
-
-\begin_layout Plain Layout
-
-    piece = floor(rand * 64) % 32
-\end_layout
-
-\begin_layout Plain Layout
-
-    dev = (rand * 64) % 1
-\end_layout
-
-\begin_layout Plain Layout
-
-    if dev >= P[piece]:
-\end_layout
-
-\begin_layout Plain Layout
-
-        result = Y[piece] + dev * Z[piece]
-\end_layout
-
-\begin_layout Plain Layout
-
-    elif piece <= 15:
-\end_layout
-
-\begin_layout Plain Layout
-
-        result = S[piece] + dev * Q[piece]
-\end_layout
-
-\begin_layout Plain Layout
-
-    elif piece < 31:
-\end_layout
-
-\begin_layout Plain Layout
-
-        piece = piece - 16
-\end_layout
-
-\begin_layout Plain Layout
-
-        while True:
-\end_layout
-
-\begin_layout Plain Layout
-
-            u,
- v = random(),
- random()
-\end_layout
-
-\begin_layout Plain Layout
-
-            if u > v:
-\end_layout
-
-\begin_layout Plain Layout
-
-                u,
- v = v,
- u
-\end_layout
-
-\begin_layout Plain Layout
-
-            result = S[piece] + u/5
-\end_layout
-
-\begin_layout Plain Layout
-
-            if v >= D[piece]:
-\end_layout
-
-\begin_layout Plain Layout
-
-                break
-\end_layout
-
-\begin_layout Plain Layout
-
-            if v < u + E[piece] * (exp((S[piece+1]**2 - abs**2)/2) - 1):
-\end_layout
-
-\begin_layout Plain Layout
-
-                break
-\end_layout
-
-\begin_layout Plain Layout
-
-    else:
-\end_layout
-
-\begin_layout Plain Layout
-
-        while True:
-\end_layout
-
-\begin_layout Plain Layout
-
-            result = sqrt(9 - 2*log(random()))
-\end_layout
-
-\begin_layout Plain Layout
-
-            if result * random() < 3:
-\end_layout
-
-\begin_layout Plain Layout
-
-                break
-\end_layout
-
-\begin_layout Plain Layout
-
-    if sign:
-\end_layout
-
-\begin_layout Plain Layout
-
-        result = -result
-\end_layout
-
-\begin_layout Plain Layout
-
-    return result
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-rexerc16[HM22]
-\end_layout
-
-\end_inset
-
-(J.
- H.
- Ahrens.) Develop an algorithm for gamma deviates of order 
-\begin_inset Formula $a$
-\end_inset
-
- when 
-\begin_inset Formula $0<a<1$
-\end_inset
-
-,
- using the rejection method with 
-\begin_inset Formula $cg(t)=t^{a-1}/\Gamma(a)$
-\end_inset
-
- for 
-\begin_inset Formula $0<t<1$
-\end_inset
-
-,
- and with 
-\begin_inset Formula $cg(t)=\text{e}^{-t}/\Gamma(a)$
-\end_inset
-
- for 
-\begin_inset Formula $t\geq1$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-The density function for the gamma distribution is
-\begin_inset Formula 
-\begin{align*}
-f(x) & \coloneqq\frac{1}{\Gamma(a)}x^{a-1}\text{e}^{-x}, & x & \geq0.
-\end{align*}
-
-\end_inset
-
-Let
-\begin_inset Formula 
-\[
-g(t)\coloneqq\begin{cases}
-t^{a-1}, & 0<t<1;\\
-\text{e}^{-t}, & t\geq1.
-\end{cases}
-\]
-
-\end_inset
-
-Then,
-\begin_inset Formula 
-\begin{align*}
-G_{0}(x)\coloneqq\int_{0}^{x}g(t)\text{d}t & =\begin{cases}
-\int_{0}^{x}t^{a-1}\text{d}t=\left[\frac{t^{a}}{a}\right]_{t=0}^{x}=\frac{x^{a}}{a}, & 0<t<1;\\
-a^{-1}+\int_{1}^{x}\text{e}^{-t}\text{d}t=\frac{1}{a}-\left[\text{e}^{-t}\right]_{t=1}^{x}=a^{-1}+\text{e}^{-1}-\text{e}^{-x}, & t\geq1.
-\end{cases}
-\end{align*}
-
-\end_inset
-
-If 
-\begin_inset Formula $M\coloneqq\lim_{x\to\infty}G(x)=a^{-1}+\text{e}^{-1}$
-\end_inset
-
-,
- the real probability distribution function is 
-\begin_inset Formula $G(x)\coloneqq G_{0}(x)/M$
-\end_inset
-
-.
- Thus,
- a possible method could be the following:
- Generate to independent deviates 
-\begin_inset Formula $U$
-\end_inset
-
- and 
-\begin_inset Formula $V$
-\end_inset
-
- uniformly distributed between 0 and 1.
- If 
-\begin_inset Formula $U<a^{-1}/M$
-\end_inset
-
-,
- set 
-\begin_inset Formula $X\coloneqq\sqrt[a]{aMU}=\sqrt[a]{1+\frac{a}{\text{e}}}\sqrt[a]{U}$
-\end_inset
-
- and 
-\begin_inset Formula $Y\coloneqq\Gamma(a)f(X)/g(X)=\text{e}^{-X}$
-\end_inset
-
-.
- Otherwise set 
-\begin_inset Formula $X\coloneqq-\ln M\cdot\ln(1-U)$
-\end_inset
-
- and 
-\begin_inset Formula $Y\coloneqq\Gamma(a)f(X)/g(X)=X^{a-1}$
-\end_inset
-
-.
- If 
-\begin_inset Formula $V\geq Y$
-\end_inset
-
-,
- reject 
-\begin_inset Formula $X$
-\end_inset
-
- and repeat all the steps.
- Otherwise return 
-\begin_inset Formula $X$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-rexerc17[M24]
-\end_layout
-
-\end_inset
-
-What is the 
-\emph on
-distribution function
-\emph default
- 
-\begin_inset Formula $F(x)$
-\end_inset
-
- for the geometric distribution with probability 
-\begin_inset Formula $p$
-\end_inset
-
-?
- What is the 
-\emph on
-generating function
-\emph default
- 
-\begin_inset Formula $G(z)$
-\end_inset
-
-?
- What are the mean and standard deviation of this distribution?
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-
-\begin_inset Formula 
-\[
-F(x)=P(X\leq x)=1-P(X>x)=1-P(X>\lfloor x\rfloor)=1-(1-p)^{\max\{\lfloor x\rfloor,0\}}.
-\]
-
-\end_inset
-
-The generating function is
-\begin_inset Formula 
-\[
-G(z)\coloneqq\sum_{k\geq1}(1-p)^{k-1}pz^{k}=zp\sum_{k\geq0}(1-p)^{k}z^{k}=\frac{zp}{1-(1-p)z}=\frac{zp}{1-qz},
-\]
-
-\end_inset
-
-where 
-\begin_inset Formula $q\coloneqq1-p$
-\end_inset
-
-.
- Then
-\begin_inset Formula 
-\begin{align*}
-G'(z) & =\frac{p-pqz+pqz}{(1-qz)^{2}}=\frac{p}{(1-qz)^{2}}, & G''(z) & =-\frac{2pq}{(1-qz)^{3}},
-\end{align*}
-
-\end_inset
-
-so the mean is
-\begin_inset Formula 
-\[
-G'(1)=\frac{p}{(1-q)^{2}}=\frac{p}{p^{2}}=\frac{1}{p},
-\]
-
-\end_inset
-
-and the variance is
-\begin_inset Formula 
-\[
-G''(1)+G'(1)-G'(1)^{2}=-\frac{2pq}{(1-q)^{3}}+\frac{1}{p}-\frac{1}{p^{2}}=\frac{2q}{p^{2}}+\frac{1}{p}-\frac{1}{p^{2}}=\frac{2q+p-1}{p^{2}}=\frac{q}{p^{2}},
-\]
-
-\end_inset
-
-so the standard deviation is 
-\begin_inset Formula $\frac{\sqrt{q}}{p}$
-\end_inset
-
-.
-\end_layout
-
-\end_body
-\end_document
-- 
cgit v1.2.3