-
Notifications
You must be signed in to change notification settings - Fork 4
/
Copy pathreadings.html
472 lines (337 loc) · 22.1 KB
/
readings.html
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
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Stat 134, Spring '23 | Readings</title>
<!-- Bootstrap -->
<link href="https://fonts.googleapis.com/css?family=Raleway:400,700|PT+Serif|Lora" rel="stylesheet">
<link href="css/bootstrap.min.css" rel="stylesheet">
<link rel="stylesheet" type="text/css" href="css/main.css"/>
<!-- Swiper -->
<link rel="stylesheet" href="css/swiper.min.css">
<!-- jQuery -->
<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.1.0/jquery.min.js"></script>
<!-- MathJax -->
<script type="text/x-mathjax-config">
MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}});
</script>
<script type="text/javascript" async
src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_CHTML">
</script>
<script src="https://use.fontawesome.com/63e7afe910.js"></script>
<!-- HTML5 shim and Respond.js for IE8 support of HTML5 elements and media queries -->
<!-- WARNING: Respond.js doesn't work if you view the page via file:// -->
<!--[if lt IE 9]>
<script src="https://oss.maxcdn.com/html5shiv/3.7.3/html5shiv.min.js"></script>
<script src="https://oss.maxcdn.com/respond/1.4.2/respond.min.js"></script>
<![endif]-->
</head>
<body>
<!-- Title -->
<div class="navbar navbar-custom">
<nav class="navbar">
<div class="container">
<div class="col-sm-3 col-xs-3" id=nav-text-left>
<div class=desktop>
<ul class="nav navbar-nav">
<li><h4><a href=index.html>Stat 134</a></h4></li>
</ul>
</div>
<div class=mobile>
<ul class="nav navbar-nav">
<li><h4><a href=index.html>134</a></h4></li>
</ul>
</div>
</div>
<div class="col-sm-9 col-xs-9" id=nav-text-right>
<ul class="nav navbar-nav navbar-right">
<li><p><a href=index.html>Index/Calendar</a></p></li>
<li><p><a href=homework.html>Homework</a></p></li>
<li><p><a href=readings.html>Readings</a></p></li>
<li><p><a href=syllabus.html>Syllabus</a></p></li>
<li><p><a href=staff.html>Staff/Section/OH</a></p></li>
</ul>
</div>
</div>
</nav>
</div>
<header class="container">
<h1 class = "header-xs">Concepts of Probability</h1>
<h5 class = "header-xs" id=subheading>Stat 134 with Adam Lucas, Spring '23.</h5>
</header>
<br>
<br>
<hr>
<br>
<header class="container">
<h3 class = "header-xs">Reading Guide</h3>
<h4>attributed to Prof. Adhikari, Fall 2017</h4>
</header>
<section class="container">
<ul class=logistics>
<li><p><b>Two philosophies</b>: If you are taking lectures notes (or can borrow notes from someone who is), read the lecture notes again before reading the text, that will make the text easier to understand. If you are not taking lecture notes, skim the reading before lecture to maximize the chance that you grok (understand deeply) the lecture. Either way you approach the reading, the examples are your friends. Get to know them, many of them are fun, and they will help you understand the theory.</p></li>
<li><p>It goes without saying that the Exercises at the end of each section are important. This page lists what you should read before you try those exercises. You can skim at first but if you have trouble with the exercises then read the section carefully before trying the exercises again.</p></li>
</ul>
</section>
<header class="container">
<h3 class = "header-xs">Chapter 1</h3>
</header>
<section class="container">
<ul class=logistics>
<li><p><b>1.1</b>: Pages 1-5.</p></li>
<li><p><b>1.2</b>: Pages 11-13. It is important to understand the figure on page 13.</p></li>
<li><p><b>1.3</b>: This is the basis of everything that follows. The table on Page 19 should be imprinted on your heart and you should thoroughly understand pages 20-25. Skip Example 4 for now. Learn to recognize the Bernoulli distribution and the uniform distribution on a finite set. In this course it is a very good idea to learn to recognize the standard distributions that crop up frequently.</p></li>
<li><p><b>1.4</b>: Try following this path: start with Example 2 on page 34. This motivates the general formula at the top of page 36. Read that formula and Example 4. Then come back to Example 3 – it's the interesting one. The multiplication rule on page 37 will come as no surprise. At this point you should be able to simply read Examples 6 and 7. Compare the general multiplication rule on page 37 with the special case when you have independence, on page 42. Then skim Examples 8 and 9. See if you can draw a tree diagram that can be used in Example 9. You should now be able to go back and do Example 4 of 1.3 without reading the solution first.</p></li>
<li><p><b>1.5</b>: Page 47 through the discussion that ends at the top of page 51.</p></li>
<li><p><b>1.6</b>: This obvious extension of the familiar multiplication rule leads to a lot of interesting stuff. I suggest you start with Example 4. Then try Examples 2 and 3 (please note the infinite outcome space - everything's nice and convergent so don't worry). And then the birthday problem in Example 5. It is useful to note that independence can be slippery when you have more than two events: see Example 8, and also look carefully at the box on page 67 – it is not enough just to check that P(ABC) = P(A)P(B)P(C).</p></li>
<li><b>The Chapter Summary is terrific - read it! </b> Pages 72-73. You can ignore the bit on Odds.</li>
</ul>
</section>
<header class="container">
<h3 class = "header-xs">Chapter 2</h3>
</header>
<section class="container">
<ul class=logistics>
<li><p><b>2.1</b>: Go through pages 79-83 very thoroughly. Then read all the contents of the box on Page 86;
follow that by learning the derivation using consecutive odds. Look carefully at the arrays of figures on Pages
87-89. Go through the captions and make sure you're following the details of what changes as <span style="font-style: italic;">n</span> gets large.</p></li>
<li><p><b>2.2</b>: Everything up to and including the box on Page 101, as well as the discussion of
“How good is the normal approximation?” on pages 103-104. I will sometimes use the terms "center"
and "spread" for "mean" and "standard deviation" respectively. Also, more formally, "location parameter"
and "scale parameter". The normal table is in Appendix 5. </p></li>
<li><p><b>2.4</b>: Nice short section, read all of it. </p></li>
<li><p><b>2.5</b>: Start on Page 124 at Sampling Without Replacement and read to the end. Again a nice
short section, but be careful when you do the problems - counting can be quite slippery.</p></li>
<li><b>Chapter Summary</b>: Read the portion entitled Binomial Probability Formula on Page 130. Then go over all of Page 131 except the Square Root Law for Independent Trials.</li>
</ul>
</section>
<header class="container">
<h3 class = "header-xs">Chapter 3</h3>
</header>
<section class="container">
<ul class=logistics>
<li><p><b>3.1</b>: Skim pages 140-152, through
the box on page 152, then read carefully anything that seems unclear. In lecture we
have done most of this material but without the context of the random
variable. Skip the technical remark starting at the bottom of page 143.
Next read the first paragraph under the heading Several Random Variables (page 153), skim the
definition and consequences of mutual independence of random variables
on page 154. Verify that the messy formula for the multinomial distribution follows
the same logic as for the binomial, and look over the discussion of symmetry
on page 156. </p></li>
<li><p><b>3.2</b>: This section is the key to
much of the rest of the course. You must go through all of it,
except the Gambling Interpretation on pages 165-166 and
Expectation and Prediction on pages 178-179. The summary box on page 181 is crucial. </p></li>
<li><p><b>3.3</b>: This is about the fundamental measure of dispersion,
and like 3.2 it must be internalized deeply. Read everything except the Skewness section on page 198.</p></li>
<li><p><b>3.4</b>: This formalizes some moves
we've been making for a while, e.g. with the Poisson distribution. But
the examples in this section are all in the context of the geometric
distribution which is the simplest of all the distributions on an
infinite set. Skim the whole section.
</p></li>
<li><p><b>3.5</b>: The Poisson is familiar as
an approximation to the binomial. Here it appears in its own
right as a distribution. Read pages 222-224, then 226-227. (We will never cover
the skew-normal approx to the Poisson.) For the
random scatter, read the assumptions in the boxes on page 229 and the
statement of the theorem in the box on page 230. Then read the
Examples 2 and 3, and the note on Thinning. You don't have to read the
proof of the Poisson Scatter Theorem on page 233.</p></li>
<li><p><b>3.6</b>: This formalizes the
symmetries that you saw in card shuffling earlier in the
class. Go straight to Examples 1 and 2 on page 240 - you will find
that you could have done them back in Chapter 2 after we talked about sampling
without replacement. The main calculation is
that of the mean and variance of the hypergeometric, pages 241-243. </p></li>
<li><b>Chapter Summary</b>:
You should understand the entire summary deeply. If you don't, spend some time
reviewing anything you feel uncomfortable with before the midterm. Don't
forget the box on page 181. And take a look at the final line, which points you to
Distribution Summaries in the back of the text. These two pages
provide an excellent summary of the important basic ideas of probability. Students refer to them
long after they have completed Stat 134.
</li>
</ul>
</section>
<header class="container">
<h3 class = "header-xs">Chapter 4</h3>
</header>
<section class="container">
<ul class=logistics>
<li><p><b>4.1</b>: Pages 260-271. Follow the examples closely. Spend some time comparing pages 262-263. Understanding the relationship between between discrete distributions and continuous distributions (defined by a density) is crucial to building a bridge between what you've learned up to now to what we're doing in Ch 4.</p></li>
<li><p><b>4.2</b>: Pages 278-290, don't care about Gamma Dist for Non-Integer Shape Parameter or anything after that. Understanding the summary on page 288-289 is important.
Example 4 on page 290 is instructive because it shows how you can use
gamma facts known from the Poisson process context in a setting where
there appears to be no Poisson process.</p></li>
<li><p><b>4.4</b>: Read the whole section (it's short) and follow the examples carefully. Try hard to wrap your head around the picture on page 305. Once you get straight what it means, the whole section makes sense.</p></li>
<li><p><b>4.5</b>: We defined the c.d.f. very
early, when we started 4.1, and used it for the standard normal distributions way before that. So skim pages 311-314, then go over
Examples 1 and 2 and notice that that we did Example 1 in class when we
talked about the uniform density. The discussion of max and min
on pages 316-318 will come as no surprise, given the similarity between the geometric and the exponential distributions. Pay close attention to the discussion of Simulation starting at the bottom of page 320 and everything to the end, it's not what I would call core material but it's pretty cool.</p></li>
<li><p><b>4.6</b>: Nice short section, read it all. Remember that identifying a beta distribution is easy – the density has to look like <span style="font-style: italic;">x</span> to a power times <span style="font-style: italic;">(1-x)</span> to another power, for <span style="font-style: italic;">x</span> between 0 and 1. The rest is just the constant that makes the density integrate to 1.</p></li>
<li><p><b>Chapter Summary</b>: Everything on pages 332-333 except the sections on hazard rates.</p></li>
<li><p><b>Note</b>: Gamma densities are sprinkled throughout the chapter, in particular in
Sections 4.2 and 4.4; the gamma constant also appears in Section 4.6.</p></li>
</ul>
</section>
<header class="container">
<h3 class = "header-xs">Chapter 5</h3>
</header>
<section class="container">
<ul class=logistics>
<li>This is the shortest chapter in the book.
It is, however, very dense, and has a lot of important results in it.
The biggest shift we make in this chapter is extending the ideas of chapter 4
to joint distributions.
</li>
<br>
<li><p><b>5.1</b>: Easy but important;
go through all the examples. This section gives you practice in
representing events as regions in the plane.</p></li>
<li><p><b>5.2</b>: This one has all the
fundamentals, so read everything carefully. The three examples aren't overly hard,
so I recommend attempting them on your own before reading the solution. It is
important to compare the tables on pages 348-349 line by line. You will
find that you already learned all the joint density facts in Chapter 3,
provided you replace sums by integrals.</p></li>
<li><p><b>5.3</b>: This is perhaps the most
important joint distribution in statistics. Read pages 357-361 (we'll do
most of it in lecture). Then read the result in the box on
page 363. The result is crucial (sums of independent normals are
normal) and simple to remember, even if you choose not to go through
its derivation. Example 2 on page 364 involves an important
technique.
<!--
Before you read about the chi-squared distribution, go
back to the discussion of the gamma function for half-integer values of
<span style="font-style: italic;">r</span>, from the bottom of page 290 through 292.
we skipped that at the time, but it's relevant now. Then read the chi-square
section from page 364 to just below equation (2) on page 365 where the
chi-squared distribution is defined.
You can ignore the rest unless you
have already taken a statistics class which covered chi-squared tests.<br>
Read the rest of this section when and if you take Stat 135.</p></li>
-->
<li><p><b>5.4</b>: We will cover
distributions of sums, pages 371-382. The ratio example is great but
everything that I ask you to do with ratios can be done without the
density of the ratio, so I have omitted the density calculation</p></li>
<li><p><b>Chapter Summary</b>: Nice and short.
<!--
, and you can skip the density of X+Y and the convolution formula
-->
At this point you should go through the Distribution Summaries (pages 476-478) and notice
that you know all the distributions, apart from the bivariate normal which you
will meet in Chapter 6. These summaries are a wonderful part of this
text; you won't find this information so succinctly displayed elsewhere.</p></li>
</ul>
</section>
<header class="container">
<h3 class = "header-xs">Chapter 6</h3>
</header>
<section class="container">
<ul class=logistics>
<li>In Ch 4 you started working with continuous variables, a bit like if you didn’t know
how to swim and I tried to explain it to you. Then in Ch 5, I threw you into the deep end.
Now in sections 1 and 2 of Ch 6 I’m dragging you out and comforting you by talking
about discrete random variables again. Make sure to catch your breath, because in
sections 3-5 you’re going back in the deep end.
<!-- The good news is that sections
3-5 will happen slowly, because they are broken up by Veteran’s day and then
Thanksgiving, but try to use those breaks to refresh yourself and hopefully solidify
your knowledge rather than pretending the course doesn’t exist and forgetting everything.
-->
We’re in the home stretch. Come to lecture and hopefully some of the intuition will
sink in. Talk to other students, come to SLC/office hours, come to section. It’s especially
hard to get the intuition for this part of the class on your own.
</li>
<br>
<li><p><b>6.1</b>: This is essentially just
one example, to get you back into thinking about discrete joint
distributions. Notice that, as with many examples in conditioning, it's
easy to find conditional distributions if you go in "chronological
order". For example, it's usually easy to find the conditional
distribution of the
number of heads (<span style="font-style: italic;">Y</span>) given the
number of coins (<span style="font-style: italic;">X</span>).
It takes
more work
to "go backwards in time", that is, to find the distribution of the
number of coins given the number of heads. You need to renormalize by
the probability of what's given; that is, you have to use the division
rule. That's what this section is
about.</p></li>
<li><p><b>6.2</b>: Conditional expectation is
a powerful tool for finding expectations. The key is the box on Page
403. Skim pages 40-403, then read Examples 2 and 3. Then go to
Page 406, which formalizes a natural idea.</p></li>
<li><p><b>6.3</b>: Start with the box on
page 417, in the context of coin-tossing. Read Example 3 and Problem 1
of Example 4. Now go over the boxes on pages 410 and 411, and then look
at the calculations at the bottom of page 411 to reassure yourself that
a conditional density is just an ordinary density, and can be used like
any other density. The diagrams on page 412, and their companion text
on page 413, are terrific for a geometric understanding of the division
involved in the formula for the conditional density. Go on to Example 1
and follow it thoroughly. Then look at the box on page 416 and go over
Example 2. Finally, compare pages 424 and 425. They should show
you that everything you know about continuous conditioning is an
extension of what you already knew about discrete conditioning.</p></li>
<li><p><b>6.4</b>: We will first use
covariance as a tool to find the variance of a sum. So start with
pages 430-431, then jump to page 441-444. Next comes correlation,
which is the measure that gives some meaning to covariance. Go over the
boxes on pages 432-433. Example 4 of the text is similar to one
of the exercises on that page. Example 6 brings together all the
techniques you have recently learned. It's well worth going through.</p></li>
<li><p><b>6.5</b>: The bivariate (and
multivariate) normal is the fundamental distribution of statistics.
Read page 449, then the box on page 451. If you don't like the geometry
of the construction of the bivariate normal, never mind (though pages
452-453 are among the best descriptions of the geometry at this level).
But you must follow everything on pages 454-461. Much of it will be
done in class exactly as in the text, but you must fill in the blanks.</p></li>
<li><p><b>Chapter Summary</b>: This lists all the general formulas, but in my experience students
understand these formulas much better in the context of specific
examples. If a formula seems mysterious, an excellent exercise is to go
through your notes and the text to find one specific example of the use
of that formula.</p></li>
</ul>
</section>
<hr>
<footer section="container">
<row class="col-sm-12">
<ul class="col-sm-12">
<li>Site originally designed and developed by <a href="https://www.stat.berkeley.edu/~ani/.com" target="_blank">Ani Adhikari</a> and <a href="https://malayandi.github.io/" target="_blank">Andy Palan</a>. </li>
<br>
</ul>
</row>
</footer>
<!-- jQuery (necessary for Bootstrap's JavaScript plugins) -->
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.12.4/jquery.min.js"></script>
<!-- Include all compiled plugins (below), or include individual files as needed -->
<script src="js/bootstrap.min.js"></script>
<!-- Swiper -->
<script src="js/swiper.min.js"></script>
<!-- Initializing Swiper -->
<script>
var mySwiper = new Swiper ('.swiper-container', {
// If we need pagination
pagination: '.swiper-pagination',
paginationClickable: true,
// Navigation arrows
nextButton: '.swiper-button-next',
prevButton: '.swiper-button-prev',
// And if we need scrollbar
scrollbar: '.swiper-scrollbar',
// Default Slide i.e. Week Number
initialSlide: 0
})
</script>
</body>
</html>