forked from gadepall/digital-communication
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathmain.aux
427 lines (427 loc) · 31.3 KB
/
main.aux
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
\relax
\providecommand\hyper@newdestlabel[2]{}
\bibstyle{plainnat}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\@writefile{toc}{\vskip -8pt}
\@writefile{toc}{\contentsline {schapter}{Introduction}{iii}{chapter*.2}\protected@file@percent }
\@writefile{toc}{\contentsline {chapter}{\numberline {1}Introduction}{1}{chapter.1}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {chapter}{\numberline {2}Axioms}{3}{chapter.2}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {2.1}Examples}{3}{section.2.1}\protected@file@percent }
\newlabel{eq:ncert/10/15/1/3/1}{{2.1}{3}{Examples}{equation.2.1.1}{}}
\newlabel{eq:ncert/10/15/1/3/2}{{2.2}{3}{Examples}{equation.2.1.2}{}}
\newlabel{eq:ncert/10/15/1/3/3}{{2.3}{3}{Examples}{equation.2.1.3}{}}
\newlabel{eq:ncert/10/15/1/3/4}{{2.4}{4}{Examples}{equation.2.1.4}{}}
\newlabel{eq:ncert/10/15/1/4/prob}{{2.6}{4}{Examples}{equation.2.1.6}{}}
\newlabel{eq:ncert/10/15/1/7/1}{{2.15}{6}{Examples}{equation.2.1.15}{}}
\@writefile{lot}{\contentsline {table}{\numberline {2.1}{\ignorespaces }}{8}{table.2.1}\protected@file@percent }
\newlabel{tab:10/15/2/4/1}{{2.1}{8}{}{table.2.1}{}}
\newlabel{eq:ncert/11/16/3/12/1}{{2.29}{9}{Examples}{equation.2.1.29}{}}
\newlabel{eq:ncert/11/16/3/12/2}{{2.31}{10}{Examples}{equation.2.1.31}{}}
\newlabel{eq:ncert/11/16/3/12/3}{{2.35}{10}{Examples}{equation.2.1.35}{}}
\gdef \LT@i {\LT@entry
{1}{112.79999pt}\LT@entry
{1}{92.4pt}\LT@entry
{1}{89.4pt}}
\@writefile{lot}{\contentsline {table}{\numberline {2.3}{\ignorespaces }}{13}{table.2.3}\protected@file@percent }
\newlabel{table:ncert/11/16/3/18/}{{2.3}{13}{}{table.2.3}{}}
\@writefile{lot}{\contentsline {table}{\numberline {2.4}{\ignorespaces Definition of $X$.}}{14}{table.2.4}\protected@file@percent }
\newlabel{tab:ncert/11/16/3/21/X-def}{{2.4}{14}{Definition of $X$}{table.2.4}{}}
\@writefile{lot}{\contentsline {table}{\numberline {2.5}{\ignorespaces Definition of $Y$.}}{14}{table.2.5}\protected@file@percent }
\newlabel{tab:ncert/11/16/3/21/Y-def}{{2.5}{14}{Definition of $Y$}{table.2.5}{}}
\newlabel{eq:ncert/11/16/3/21/pr-X}{{2.60}{15}{Examples}{equation.2.1.60}{}}
\newlabel{eq:ncert/11/16/3/21/pr-Y}{{2.61}{15}{Examples}{equation.2.1.61}{}}
\newlabel{eq:ncert/11/16/3/21/pr-XY}{{2.62}{15}{Examples}{equation.2.1.62}{}}
\@writefile{lot}{\contentsline {table}{\numberline {2.6}{\ignorespaces Joint pmf of $X$ and $Y$.}}{15}{table.2.6}\protected@file@percent }
\newlabel{tab:ncert/11/16/3/21/joint}{{2.6}{15}{Joint pmf of $X$ and $Y$}{table.2.6}{}}
\@writefile{lot}{\contentsline {table}{\numberline {2.7}{\ignorespaces Random variable X}}{16}{table.2.7}\protected@file@percent }
\newlabel{tab:11/16/4/3/1}{{2.7}{16}{Random variable X}{table.2.7}{}}
\newlabel{eq:ncert/11/16/4/7/test}{{2.91}{18}{Examples}{equation.2.1.91}{}}
\newlabel{eq:ncert/12/13/2/9/1}{{2.117}{22}{Examples}{equation.2.1.117}{}}
\newlabel{eq:ncert/12/13/2/9/7}{{2.119}{22}{Examples}{equation.2.1.119}{}}
\newlabel{ncert/12/13/2/4}{{{2.1.28}}{23}{Examples}{Item.92}{}}
\newlabel{ncert/12/13/2/5}{{{2.1.29}}{23}{Examples}{Item.93}{}}
\newlabel{ncert/12/13/2/6}{{{2.1.31}}{23}{Examples}{Item.95}{}}
\newlabel{ncert/12/13/2/8}{{{2.1.32}}{23}{Examples}{Item.98}{}}
\newlabel{ncert/12/13/2/9}{{{2.1.33}}{23}{Examples}{Item.99}{}}
\newlabel{ncert/12/13/2/10}{{{2.1.34}}{24}{Examples}{Item.100}{}}
\newlabel{ncert/12/13/2/13}{{{2.1.35}}{24}{Examples}{Item.105}{}}
\newlabel{ncert/12/13/2/14}{{{2.1.36}}{25}{Examples}{Item.110}{}}
\newlabel{it:ncert/12/13/2/18/}{{{2.1.38}a}{27}{Examples}{Item.123}{}}
\newlabel{tab:ncert/12/13/6/13/1}{{{2.1.42}}{29}{Examples}{Item.134}{}}
\@writefile{lot}{\contentsline {table}{\numberline {2.8}{\ignorespaces Given Information}}{29}{table.2.8}\protected@file@percent }
\newlabel{eq:ncert/12/13/6/13/4}{{2.153}{29}{Examples}{equation.2.1.153}{}}
\newlabel{eq:ncert/12/13/6/13/5}{{2.154}{29}{Examples}{equation.2.1.154}{}}
\newlabel{eq:ncert/12/13/6/15/tot}{{2.159}{30}{Examples}{equation.2.1.159}{}}
\newlabel{eq:ncert/12/13/6/15/bot}{{2.160}{30}{Examples}{equation.2.1.160}{}}
\newlabel{eq:ncert/12/13/6/15/cot}{{2.161}{30}{Examples}{equation.2.1.161}{}}
\newlabel{eq:ncert/12/13/6/15/dot}{{2.162}{30}{Examples}{equation.2.1.162}{}}
\newlabel{eq:ncert/12/13/6/15/1}{{2.164}{30}{Examples}{equation.2.1.164}{}}
\newlabel{eq:ncert/12/13/6/15/alpha}{{2.165}{30}{Examples}{equation.2.1.165}{}}
\newlabel{eq:ncert/12/13/6/15/tat}{{2.168}{30}{Examples}{equation.2.1.168}{}}
\newlabel{eq:ncert/12/13/6/18/c}{{{2.1.45}c}{31}{Examples}{Item.148}{}}
\newlabel{prob:12/13/6/19}{{{2.1.46}b}{32}{Examples}{Item.152}{}}
\gdef \LT@ii {\LT@entry
{1}{144.94148pt}\LT@entry
{1}{144.54149pt}\LT@entry
{1}{144.54149pt}}
\@writefile{toc}{\contentsline {section}{\numberline {2.2}Exercises}{33}{section.2.2}\protected@file@percent }
\@writefile{lot}{\contentsline {table}{\numberline {2.10}{\ignorespaces }}{34}{table.2.10}\protected@file@percent }
\newlabel{tab:exemplar/11/16/3/8/table2}{{2.10}{34}{}{table.2.10}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.1}{\ignorespaces Question Figure}}{36}{figure.2.1}\protected@file@percent }
\newlabel{fig:exemplar/11/16/3/11/Venn_Diagram}{{2.1}{36}{Question Figure}{figure.2.1}{}}
\newlabel{eq:exemplar/11/16/3/11/1}{{2.191}{36}{Exercises}{equation.2.2.191}{}}
\newlabel{eq:exemplar/11/16/3/11/2}{{2.192}{36}{Exercises}{equation.2.2.192}{}}
\newlabel{eq:exemplar/11/16/3/11/8}{{2.193}{36}{Exercises}{equation.2.2.193}{}}
\newlabel{eq:exemplar/11/16/3/11/4}{{2.194}{36}{Exercises}{equation.2.2.194}{}}
\newlabel{eq:exemplar/11/16/3/11/6}{{2.196}{36}{Exercises}{equation.2.2.196}{}}
\newlabel{eq:exemplar/11/16/3/11/5}{{2.197}{36}{Exercises}{equation.2.2.197}{}}
\newlabel{eq:exemplar/11/16/3/11/7}{{2.198}{36}{Exercises}{equation.2.2.198}{}}
\newlabel{eq:exemplar/11/16/3/11/axiom_sum_A}{{2.200}{37}{Exercises}{equation.2.2.200}{}}
\newlabel{eq:exemplar/11/16/3/11/3}{{2.203}{37}{Exercises}{equation.2.2.203}{}}
\newlabel{Axioms}{{2.205}{37}{Exercises}{equation.2.2.205}{}}
\gdef \LT@iii {\LT@entry
{1}{62.79999pt}\LT@entry
{1}{222.4pt}}
\@writefile{toc}{\contentsline {chapter}{\numberline {3}Random Variables}{39}{chapter.3}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{lot}{\contentsline {table}{\numberline {3.2}{\ignorespaces }}{39}{table.3.2}\protected@file@percent }
\newlabel{tab:ncert/10/15/1/15/Table1}{{3.2}{39}{}{table.3.2}{}}
\gdef \LT@iv {\LT@entry
{1}{42.79999pt}\LT@entry
{1}{92.4pt}\LT@entry
{1}{262.4pt}}
\gdef \LT@v {\LT@entry
{1}{65.79999pt}\LT@entry
{1}{65.4pt}\LT@entry
{1}{187.4pt}\LT@entry
{1}{65.0pt}}
\@writefile{lot}{\contentsline {table}{\numberline {3.4}{\ignorespaces Random variables(RV) X,Y and X,Y}}{41}{table.3.4}\protected@file@percent }
\newlabel{table1:11/16/4/2}{{3.4}{41}{Random variables(RV) X,Y and X,Y}{table.3.4}{}}
\@writefile{lot}{\contentsline {table}{\numberline {3.6}{\ignorespaces }}{41}{table.3.6}\protected@file@percent }
\newlabel{tab:11/16/4/4}{{3.6}{41}{}{table.3.6}{}}
\gdef \LT@vi {\LT@entry
{1}{39.79999pt}\LT@entry
{1}{62.4pt}\LT@entry
{1}{152.4pt}}
\@writefile{lot}{\contentsline {table}{\numberline {3.8}{\ignorespaces }}{43}{table.3.8}\protected@file@percent }
\newlabel{tab:11/16//5/1}{{3.8}{43}{}{table.3.8}{}}
\gdef \LT@vii {\LT@entry
{1}{45.79999pt}\LT@entry
{1}{76.4pt}\LT@entry
{1}{152.4pt}}
\@writefile{lot}{\contentsline {table}{\numberline {3.10}{\ignorespaces }}{44}{table.3.10}\protected@file@percent }
\newlabel{tab:12/13/2/2/}{{3.10}{44}{}{table.3.10}{}}
\newlabel{ncert/12/13/2/3/defs.tex}{{{3.8}}{44}{Random Variables}{Item.199}{}}
\newlabel{ncert/12/13/2/12}{{{3.9}}{45}{Random Variables}{Item.200}{}}
\newlabel{ncert/12/13/2/15}{{{3.10}}{45}{Random Variables}{Item.204}{}}
\@writefile{lot}{\contentsline {table}{\numberline {3.11}{\ignorespaces }}{46}{table.3.11}\protected@file@percent }
\newlabel{tab:ncert/12/13/2/17/xxx}{{3.11}{46}{}{table.3.11}{}}
\@writefile{lot}{\contentsline {table}{\numberline {3.12}{\ignorespaces Random Variable Declaration}}{46}{table.3.12}\protected@file@percent }
\newlabel{tab:ncert/12/13/3/2/1}{{3.12}{46}{Random Variable Declaration}{table.3.12}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {4}Conditional Probability}{49}{chapter.4}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{eq:12/13/1/3/1}{{4.7}{50}{Conditional Probability}{equation.4.0.7}{}}
\newlabel{eq:12/13/1/3/2}{{4.12}{51}{Conditional Probability}{equation.4.0.12}{}}
\newlabel{eq:1}{{4.15}{52}{Conditional Probability}{equation.4.0.15}{}}
\newlabel{eq:2}{{4.17}{52}{Conditional Probability}{equation.4.0.17}{}}
\newlabel{eq:3}{{4.19}{52}{Conditional Probability}{equation.4.0.19}{}}
\gdef \LT@viii {\LT@entry
{1}{82.79999pt}\LT@entry
{1}{93.4pt}}
\@writefile{lot}{\contentsline {table}{\numberline {4.2}{\ignorespaces Probability of Events.}}{53}{table.4.2}\protected@file@percent }
\newlabel{table:}{{4.2}{53}{Probability of Events}{table.4.2}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.3}{\ignorespaces }}{55}{table.4.3}\protected@file@percent }
\newlabel{tab:ncert/12/13/1/13/table1}{{4.3}{55}{}{table.4.3}{}}
\newlabel{eq:12/13/1/13/eq7}{{4.42}{55}{Conditional Probability}{equation.4.0.42}{}}
\newlabel{eq:12/13/1/13/eq2}{{4.43}{55}{Conditional Probability}{equation.4.0.43}{}}
\newlabel{eq:12/13/1/13/eq5}{{4.44}{56}{Conditional Probability}{equation.4.0.44}{}}
\newlabel{eq:12/13/1/13/eq6}{{4.46}{56}{Conditional Probability}{equation.4.0.46}{}}
\newlabel{eq:12/13/1/13/eq4}{{4.47}{56}{Conditional Probability}{equation.4.0.47}{}}
\newlabel{eq:12/13/1/13/eq9}{{4.50}{56}{Conditional Probability}{equation.4.0.50}{}}
\newlabel{eq:12/13/1/13/eq8}{{4.51}{56}{Conditional Probability}{equation.4.0.51}{}}
\newlabel{eq:12/13/1/13/eq10}{{4.53}{56}{Conditional Probability}{equation.4.0.53}{}}
\newlabel{eq:12/13/1/13/eq1}{{4.54}{56}{Conditional Probability}{equation.4.0.54}{}}
\newlabel{eq:ncert/12/13/1/17/cond}{{4.59}{57}{Conditional Probability}{equation.4.0.59}{}}
\newlabel{eq:ncert/12/13/1/17/bayes-1}{{4.60}{58}{Conditional Probability}{equation.4.0.60}{}}
\newlabel{eq:ncert/12/13/1/17/bayes-2}{{4.61}{58}{Conditional Probability}{equation.4.0.61}{}}
\newlabel{eq:ncert/12/13/1/17/a-b}{{4.62}{58}{Conditional Probability}{equation.4.0.62}{}}
\newlabel{eq:ncert/12/13/1/17/great}{{4.65}{58}{Conditional Probability}{equation.4.0.65}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.4}{\ignorespaces Joint pmf for events $A$ and $B$.}}{59}{table.4.4}\protected@file@percent }
\newlabel{tab:ncert/12/13/1/17/joint}{{4.4}{59}{Joint pmf for events $A$ and $B$}{table.4.4}{}}
\newlabel{eq:ncert/12/13/1/15/pmf-X}{{4.68}{59}{Conditional Probability}{equation.4.0.68}{}}
\newlabel{eq:ncert/12/13/1/15/pmf-Y}{{4.69}{59}{Conditional Probability}{equation.4.0.69}{}}
\newlabel{eq:ncert/12/13/1/15/pmf-XY}{{4.70}{60}{Conditional Probability}{equation.4.0.70}{}}
\newlabel{ncert/12/13/2/7}{{{4.14}}{60}{Conditional Probability}{Item.261}{}}
\newlabel{ncert/12/13/3/1}{{{4.15}}{61}{Conditional Probability}{Item.266}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.5}{\ignorespaces Random variables X,Y}}{61}{table.4.5}\protected@file@percent }
\newlabel{tab:12/13/3/1/1}{{4.5}{61}{Random variables X,Y}{table.4.5}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.6}{\ignorespaces Probabilities}}{61}{table.4.6}\protected@file@percent }
\newlabel{tab:12/13/3/1/2}{{4.6}{61}{Probabilities}{table.4.6}{}}
\newlabel{ncert/12/13/3/2}{{{4.16}}{61}{Conditional Probability}{Item.267}{}}
\newlabel{ncert/12/13/3/3}{{{4.17}}{61}{Conditional Probability}{Item.268}{}}
\newlabel{tab:ncert/12/13/3/4/1}{{{4.18}}{63}{Conditional Probability}{Item.269}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.7}{\ignorespaces Random Variable and their description}}{63}{table.4.7}\protected@file@percent }
\newlabel{tab:ncert/12/13/3/4/2}{{{4.18}}{63}{Conditional Probability}{table.4.7}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.8}{\ignorespaces Probability of events}}{63}{table.4.8}\protected@file@percent }
\newlabel{tab:ncert/12/13/3/5/1}{{{4.19}}{64}{Conditional Probability}{equation.4.0.86}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.9}{\ignorespaces Given Information}}{64}{table.4.9}\protected@file@percent }
\@writefile{lot}{\contentsline {table}{\numberline {4.10}{\ignorespaces Definition of $X$.}}{64}{table.4.10}\protected@file@percent }
\newlabel{tab:ncert/12/13/3/6/X-def}{{4.10}{64}{Definition of $X$}{table.4.10}{}}
\newlabel{eq:ncert/12/13/3/6/X-pmf}{{4.89}{65}{Conditional Probability}{equation.4.0.89}{}}
\newlabel{eq:ncert/12/13/3/6/Y-1-def}{{4.90}{65}{Conditional Probability}{equation.4.0.90}{}}
\newlabel{eq:ncert/12/13/3/6/Y-2-def}{{4.91}{65}{Conditional Probability}{equation.4.0.91}{}}
\newlabel{eq:ncert/12/13/3/6/Y-3-def}{{4.92}{65}{Conditional Probability}{equation.4.0.92}{}}
\newlabel{eq:ncert/12/13/3/6/Y-def}{{4.93}{65}{Conditional Probability}{equation.4.0.93}{}}
\newlabel{eq:ncert/12/13/3/6/e1}{{4.95}{65}{Conditional Probability}{equation.4.0.95}{}}
\newlabel{eq:ncert/12/13/3/6/e2}{{4.96}{65}{Conditional Probability}{equation.4.0.96}{}}
\gdef \LT@ix {\LT@entry
{1}{64.91563pt}\LT@entry
{1}{57.6808pt}\LT@entry
{1}{311.4242pt}}
\newlabel{eq:ncert/12/13/3/6/pr-Y1}{{4.99}{66}{Conditional Probability}{equation.4.0.99}{}}
\gdef \LT@x {\LT@entry
{1}{112.79999pt}\LT@entry
{1}{94.4pt}\LT@entry
{1}{163.4pt}}
\@writefile{lot}{\contentsline {table}{\numberline {4.12}{\ignorespaces Random variables(RV) X,Y}}{67}{table.4.12}\protected@file@percent }
\newlabel{tab:ncert/12/13/3/9/1}{{4.12}{67}{Random variables(RV) X,Y}{table.4.12}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.14}{\ignorespaces Probabilities}}{67}{table.4.14}\protected@file@percent }
\newlabel{tab:ncert/12/13/3/9/2}{{4.14}{67}{Probabilities}{table.4.14}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.15}{\ignorespaces Random variables $A,X$}}{68}{table.4.15}\protected@file@percent }
\newlabel{tab:12/13/3/13/1}{{4.15}{68}{Random variables $A,X$}{table.4.15}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.16}{\ignorespaces Probabilities}}{68}{table.4.16}\protected@file@percent }
\newlabel{tab:12/13/3/13/2}{{4.16}{68}{Probabilities}{table.4.16}{}}
\newlabel{eq:12/13/3/143.0.1}{{4.107}{69}{Conditional Probability}{equation.4.0.107}{}}
\newlabel{eq:ncert/12/13/6/1/1}{{4.112}{70}{Conditional Probability}{equation.4.0.112}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.17}{\ignorespaces Random variables $X$}}{71}{table.4.17}\protected@file@percent }
\newlabel{tab:ncert/12/13/6/2/1}{{4.17}{71}{Random variables $X$}{table.4.17}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.18}{\ignorespaces Probabilities}}{71}{table.4.18}\protected@file@percent }
\newlabel{tab:ncert/12/13/6/2/2}{{4.18}{71}{Probabilities}{table.4.18}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.19}{\ignorespaces }}{73}{table.4.19}\protected@file@percent }
\newlabel{tab:ncert/12/13/6/3/table1}{{4.19}{73}{}{table.4.19}{}}
\newlabel{eq:ncert/12/13/6/3/eq1}{{4.128}{73}{Conditional Probability}{equation.4.0.128}{}}
\newlabel{eq:ncert/12/13/6/3/eq3}{{4.133}{74}{Conditional Probability}{equation.4.0.133}{}}
\newlabel{eq:ncert/12/13/6/3/eq4}{{4.134}{74}{Conditional Probability}{equation.4.0.134}{}}
\newlabel{eq:ncert/12/13/6/3/eq6}{{4.137}{74}{Conditional Probability}{equation.4.0.137}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.20}{\ignorespaces Question Table}}{75}{table.4.20}\protected@file@percent }
\newlabel{tab:12/13/6/12/table1}{{4.20}{75}{Question Table}{table.4.20}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.21}{\ignorespaces Events Table}}{75}{table.4.21}\protected@file@percent }
\newlabel{tab:12/13/6/12/table2}{{4.21}{75}{Events Table}{table.4.21}{}}
\newlabel{tab:ncert/12/13/6/16/1}{{{4.33}}{77}{Conditional Probability}{equation.4.0.151}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4.22}{\ignorespaces Final probabilities of the events.}}{77}{table.4.22}\protected@file@percent }
\@writefile{toc}{\contentsline {chapter}{\numberline {5}Distributions}{79}{chapter.5}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {5.1}Bernoulli}{79}{section.5.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {5.2}Multinomial}{80}{section.5.2}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {5.1}{\ignorespaces Spinner}}{82}{figure.5.1}\protected@file@percent }
\newlabel{fig:ncert/10/15/1/12/my_label}{{5.1}{82}{Spinner}{figure.5.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.2}{\ignorespaces Plot of Probability Mass Function}}{84}{figure.5.2}\protected@file@percent }
\newlabel{fig:ncert/10/15/1/12/pmf}{{5.2}{84}{Plot of Probability Mass Function}{figure.5.2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.3}{\ignorespaces Plot of Cumulative Distribution Function}}{85}{figure.5.3}\protected@file@percent }
\newlabel{fig:ncert/10/15/1/12/cdf}{{5.3}{85}{Plot of Cumulative Distribution Function}{figure.5.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.3}Uniform}{87}{section.5.3}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {5.4}Binomial}{89}{section.5.4}\protected@file@percent }
\newlabel{eq:ncert/11/16/3/8/pmf-X}{{5.39}{89}{Binomial}{equation.5.4.39}{}}
\newlabel{eq:ncert/11/16/3/8/def-Y}{{5.40}{90}{Binomial}{equation.5.4.40}{}}
\newlabel{eq:ncert/11/16/3/8/pmf-Y}{{5.41}{90}{Binomial}{equation.5.4.41}{}}
\newlabel{eq:ncert/11/16/3/8/cdf-Y}{{5.43}{90}{Binomial}{equation.5.4.43}{}}
\newlabel{eq:ncert/11/16/3/8/ans-i}{{5.46}{90}{Binomial}{equation.5.4.46}{}}
\newlabel{eq:ncert/11/16/3/8/ans-v}{{5.56}{91}{Binomial}{equation.5.4.56}{}}
\@writefile{lot}{\contentsline {table}{\numberline {5.1}{\ignorespaces Random variables $X_1, X_2, X_3, X$}}{93}{table.5.1}\protected@file@percent }
\newlabel{tab:ncert/12/13/1/6/1}{{5.1}{93}{Random variables $X_1, X_2, X_3, X$}{table.5.1}{}}
\newlabel{eq:ncert/12/13/1/6/2}{{5.66}{93}{Binomial}{equation.5.4.66}{}}
\newlabel{eq:ncert/12/13/1/6/3}{{5.78}{95}{Binomial}{equation.5.4.78}{}}
\newlabel{eq:ncert/12/13/1/6/4}{{5.79}{95}{Binomial}{equation.5.4.79}{}}
\newlabel{eq:ncert/12/13/1/6/5}{{5.84}{95}{Binomial}{equation.5.4.84}{}}
\newlabel{eq:ncert/12/13/1/6/1}{{5.89}{96}{Binomial}{equation.5.4.89}{}}
\gdef \LT@xi {\LT@entry
{1}{65.79999pt}\LT@entry
{1}{65.4pt}\LT@entry
{1}{219.4pt}}
\@writefile{lot}{\contentsline {table}{\numberline {5.3}{\ignorespaces Variable Description }}{97}{table.5.3}\protected@file@percent }
\newlabel{tab:13.4.4.4}{{5.3}{97}{Variable Description}{table.5.3}{}}
\newlabel{eq:ncert/12/13/4/5/2}{{5.104}{98}{Binomial}{equation.5.4.104}{}}
\newlabel{eq:ncert/12/13/5/5/10}{{5.121}{102}{Binomial}{equation.5.4.121}{}}
\newlabel{eq:ncert/12/13/5/5/14}{{5.124}{102}{Binomial}{equation.5.4.124}{}}
\newlabel{eq:ncert/12/13/5/5/16}{{5.125}{102}{Binomial}{equation.5.4.125}{}}
\@writefile{lot}{\contentsline {table}{\numberline {5.4}{\ignorespaces Given Information}}{105}{table.5.4}\protected@file@percent }
\newlabel{tab:ncert/12/13/6/5/ncert/12/13/6/5/1}{{5.4}{105}{Given Information}{table.5.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.5}Triangular}{107}{section.5.5}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {5.6}Miscellaneous}{108}{section.5.6}\protected@file@percent }
\gdef \LT@xii {\LT@entry
{1}{52.79999pt}\LT@entry
{1}{56.4pt}\LT@entry
{1}{55.4pt}\LT@entry
{1}{48.4pt}}
\gdef \LT@xiii {\LT@entry
{1}{39.79999pt}\LT@entry
{1}{48.4pt}\LT@entry
{1}{45.4pt}\LT@entry
{1}{41.4pt}\LT@entry
{1}{45.4pt}\LT@entry
{1}{43.4pt}}
\gdef \LT@xiv {\LT@entry
{1}{52.79999pt}\LT@entry
{1}{52.4pt}\LT@entry
{1}{49.4pt}\LT@entry
{1}{45.4pt}}
\gdef \LT@xv {\LT@entry
{1}{42.79999pt}\LT@entry
{1}{40.4pt}\LT@entry
{1}{45.4pt}\LT@entry
{1}{43.4pt}\LT@entry
{1}{43.4pt}\LT@entry
{1}{52.4pt}}
\gdef \LT@xvi {\LT@entry
{1}{48.58pt}\LT@entry
{1}{48.18001pt}\LT@entry
{1}{48.18001pt}\LT@entry
{1}{48.18001pt}\LT@entry
{1}{48.18001pt}\LT@entry
{1}{48.18001pt}\LT@entry
{1}{48.18001pt}\LT@entry
{1}{48.18001pt}\LT@entry
{1}{48.18001pt}}
\@writefile{toc}{\contentsline {section}{\numberline {5.7}Gaussian}{111}{section.5.7}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {5.8}Exercises}{112}{section.5.8}\protected@file@percent }
\@writefile{toc}{\contentsline {chapter}{\numberline {6}Moments}{113}{chapter.6}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {chapter}{\numberline {7}Random Algebra}{115}{chapter.7}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{lof}{\contentsline {figure}{\numberline {7.1}{\ignorespaces pmf of the sum when coin and die are rolled simultaneously}}{116}{figure.7.1}\protected@file@percent }
\newlabel{fig:ncert/11/16/3/5/pmf}{{7.1}{116}{pmf of the sum when coin and die are rolled simultaneously}{figure.7.1}{}}
\newlabel{eq:12/13/1/10/cdf}{{7.6}{117}{Random Algebra}{equation.7.0.6}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.2}{\ignorespaces $X+Y = 8 | Y < 4$}}{119}{figure.7.2}\protected@file@percent }
\newlabel{fig:ncert/12/13/1/10/1}{{7.2}{119}{$X+Y = 8 | Y < 4$}{figure.7.2}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {8}Random Numbers}{121}{chapter.8}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {8.1}Uniform Random Numbers}{121}{section.8.1}\protected@file@percent }
\@writefile{lol}{\contentsline {lstlisting}{ codes/exrand.c}{121}{lstlisting.8.-1}\protected@file@percent }
\@writefile{lol}{\contentsline {lstlisting}{ codes/coeffs.h}{122}{lstlisting.8.-2}\protected@file@percent }
\@writefile{lol}{\contentsline {lstlisting}{ codes/cdf\textunderscore plot.py }{125}{lstlisting.8.-3}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {8.1}{\ignorespaces The CDF of $U$}}{127}{figure.8.1}\protected@file@percent }
\newlabel{fig:uni_cdf}{{8.1}{127}{The CDF of $U$}{figure.8.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {8.2}Central Limit Theorem}{128}{section.8.2}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {8.2}{\ignorespaces The CDF of $X$}}{129}{figure.8.2}\protected@file@percent }
\newlabel{fig:gauss_cdf}{{8.2}{129}{The CDF of $X$}{figure.8.2}{}}
\@writefile{lol}{\contentsline {lstlisting}{ codes/pdf\textunderscore plot.py }{130}{lstlisting.8.-4}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {8.3}From Uniform to Other}{132}{section.8.3}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {8.3}{\ignorespaces The PDF of $X$}}{133}{figure.8.3}\protected@file@percent }
\newlabel{fig:gauss_pdf}{{8.3}{133}{The PDF of $X$}{figure.8.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {8.4}Triangular Distribution}{134}{section.8.4}\protected@file@percent }
\@writefile{toc}{\contentsline {chapter}{\numberline {9}Maximum Likelihood Detection: BPSK}{135}{chapter.9}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {9.1}Maximum Likelihood}{135}{section.9.1}\protected@file@percent }
\newlabel{ml-ch4_sim}{{{9.1.5}}{135}{Maximum Likelihood}{Item.504}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {10}Transformation of Random Variables}{137}{chapter.10}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {10.1}Gaussian to Other}{137}{section.10.1}\protected@file@percent }
\newlabel{ch3_raleigh_sim}{{{10.1.3}}{137}{Gaussian to Other}{Item.512}{}}
\@writefile{toc}{\contentsline {section}{\numberline {10.2}Conditional Probability}{138}{section.10.2}\protected@file@percent }
\newlabel{ch4_sim}{{{10.2.1}}{138}{Conditional Probability}{Item.513}{}}
\newlabel{ch4_anal}{{{10.2.3}}{138}{Conditional Probability}{Item.515}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {11}Bivariate Random Variables: FSK}{139}{chapter.11}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {11.1}Two Dimensions}{139}{section.11.1}\protected@file@percent }
\newlabel{ch5_fsk}{{{11.1.1}}{139}{Two Dimensions}{Item.517}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {12}Exercises}{141}{chapter.12}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {12.1}BPSK}{141}{section.12.1}\protected@file@percent }
\newlabel{prob:bpsk_decision}{{2}{141}{BPSK}{Item.522}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.1}{\ignorespaces }}{142}{figure.12.1}\protected@file@percent }
\newlabel{fig:bpsk_const}{{12.1}{142}{}{figure.12.1}{}}
\newlabel{eq:bpsk_proof_n0}{{12.5}{142}{BPSK}{equation.12.1.5}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.2}{\ignorespaces }}{144}{figure.12.2}\protected@file@percent }
\newlabel{fig:bpsk_ber}{{12.2}{144}{}{figure.12.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {12.2}Coherent BFSK}{144}{section.12.2}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {12.3}{\ignorespaces }}{145}{figure.12.3}\protected@file@percent }
\newlabel{fig:bfsk_const}{{12.3}{145}{}{figure.12.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {12.3}QPSK}{146}{section.12.3}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {12.4}$M$-PSK}{147}{section.12.4}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {12.5}Noncoherent BFSK}{148}{section.12.5}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {12.6}Craig's Formula and MGF}{152}{section.12.6}\protected@file@percent }
\newlabel{ch4_pe_mgf}{{12.55}{153}{Craig's Formula and MGF}{equation.12.6.55}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {A} Axioms }{155}{appendix.A}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {A.1}Definitions}{155}{section.A.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {A.2}Boolean Logic}{156}{section.A.2}\protected@file@percent }
\newlabel{eq:axiom_sum_A}{{{A.2.4}.1}{156}{Boolean Logic}{equation.582.1}{}}
\newlabel{eq:axiom_sum_AB0}{{{A.2.4}.2}{156}{Boolean Logic}{equation.582.2}{}}
\newlabel{eq:axiom_occurrence_of_exactly_one}{{{A.2.5}.3}{157}{Boolean Logic}{equation.583.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {A.3}Properties}{158}{section.A.3}\protected@file@percent }
\newlabel{eq:axiom_sum_one}{{{A.3.2}.2}{158}{Properties}{equation.585.2}{}}
\newlabel{eq:axiom_sum_two}{{{A.3.3}.2}{158}{Properties}{equation.586.2}{}}
\newlabel{eq:axiom_sum_ABp}{{{A.3.4}.2}{158}{Properties}{equation.587.2}{}}
\newlabel{eq:axiom_sum_AB}{{{A.3.5}.1}{158}{Properties}{equation.588.1}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {B} Z-transform}{159}{appendix.B}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{eq:dice_xz}{{{B.1}.1}{159}{Z-transform}{equation.589.1}{}}
\newlabel{eq:dice_xzprod_def}{{{B.2}.2}{159}{Z-transform}{equation.590.2}{}}
\newlabel{eq:mgf-mean}{{{B.4}.1}{160}{Z-transform}{equation.592.1}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {C}Distributions}{161}{appendix.C}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {C.1}Bernoulli }{161}{section.C.1}\protected@file@percent }
\@writefile{lol}{\contentsline {lstlisting}{ codes/bernoulli/coin.py }{161}{lstlisting.C.-6}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {C.2}Binomial Distribution}{163}{section.C.2}\protected@file@percent }
\newlabel{eq:mgf-binom}{{{C.2.2}.1}{163}{Binomial Distribution}{equation.598.1}{}}
\newlabel{eq:mean-binom}{{{C.2.3}.1}{163}{Binomial Distribution}{equation.599.1}{}}
\@writefile{lol}{\contentsline {lstlisting}{codes/binomial/binomial.py}{164}{lstlisting.C.-7}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {C.3}Uniform Distribution}{166}{section.C.3}\protected@file@percent }
\newlabel{eq:dice_pmf_xi}{{{C.3.1}.1}{166}{Uniform Distribution}{equation.601.1}{}}
\newlabel{eq:dice_xiz}{{{C.3.2}.1}{166}{Uniform Distribution}{equation.602.1}{}}
\newlabel{eq:dice_xiF}{{{C.3.3}.1}{167}{Uniform Distribution}{equation.603.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {{C.3.3}.1}{\ignorespaces CDF}}{167}{figure.603.1}\protected@file@percent }
\newlabel{fig:ncert/11/16/3/3/1}{{{C.3.3}.1}{167}{CDF}{figure.603.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {C.4}Triangular Distribution}{168}{section.C.4}\protected@file@percent }
\newlabel{eq:dice_xdef}{{{C.4.1}.1}{168}{Triangular Distribution}{equation.604.1}{}}
\newlabel{eq:dice_wrong}{{{C.4.1}.3}{168}{Triangular Distribution}{equation.604.3}{}}
\newlabel{eq:dice_x_sum}{{{C.4.2}.2}{168}{Triangular Distribution}{equation.605.2}{}}
\newlabel{eq:dice_x1_indep}{{{C.4.2}.3}{168}{Triangular Distribution}{equation.605.3}{}}
\newlabel{eq:dice_x_conv}{{{C.4.2}.4}{168}{Triangular Distribution}{equation.605.4}{}}
\newlabel{eq:dice_x_conv_x1}{{{C.4.2}.5}{169}{Triangular Distribution}{equation.605.5}{}}
\newlabel{eq:dice_x_conv_cond}{{{C.4.2}.7}{169}{Triangular Distribution}{equation.605.7}{}}
\newlabel{eq:dice_x_conv_final}{{{C.4.2}.8}{169}{Triangular Distribution}{equation.605.8}{}}
\newlabel{eq:dice_xzprod}{{{C.4.3}.2}{170}{Triangular Distribution}{equation.606.2}{}}
\newlabel{eq:dice_xz_closed}{{{C.4.3}.5}{170}{Triangular Distribution}{equation.606.5}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {{C.4.4}.1}{\ignorespaces Plot of $p_X(n)$. Simulations are close to the analysis. }}{171}{figure.607.1}\protected@file@percent }
\newlabel{fig:dice}{{{C.4.4}.1}{171}{Plot of $p_X(n)$. Simulations are close to the analysis}{figure.607.1}{}}
\@writefile{lol}{\contentsline {lstlisting}{codes/sum/dice.py}{171}{lstlisting.C.-8}\protected@file@percent }
\@writefile{toc}{\contentsline {chapter}{\numberline {D}Identities}{175}{appendix.D}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{eq:ncert/11/16/4/1/1}{{{D.1}.2}{175}{Identities}{equation.609.2}{}}
\newlabel{eq:ncert/11/16/4/1/5}{{{D.2}.1}{176}{Identities}{equation.610.1}{}}
\newlabel{eq:ncert/11/16/4/1/4}{{{D.2}.2}{176}{Identities}{equation.610.2}{}}
\gdef \@abspage@last{184}