1 00:00:00,930 --> 00:00:04,050 Welcome to the first lecture of this class. 2 00:00:04,050 --> 00:00:06,860 You may be used to having a first lecture devoted to 3 00:00:06,860 --> 00:00:09,670 general comments and motivating examples. 4 00:00:09,670 --> 00:00:11,020 This one will be different. 5 00:00:11,020 --> 00:00:14,500 We will dive into the heart of the subject right away. 6 00:00:14,500 --> 00:00:16,920 In fact, today we will accomplish a lot. 7 00:00:16,920 --> 00:00:20,030 By the end of this lecture, you will know about all of the 8 00:00:20,030 --> 00:00:23,120 elements of a probabilistic model. 9 00:00:23,120 --> 00:00:26,060 A probabilistic model is a quantitative description of a 10 00:00:26,060 --> 00:00:29,080 situation, a phenomenon, or an experiment 11 00:00:29,080 --> 00:00:31,320 whose outcome is uncertain. 12 00:00:31,320 --> 00:00:34,850 Putting together such a model involves two key steps. 13 00:00:34,850 --> 00:00:38,140 First, we need to describe the possible outcomes of the 14 00:00:38,140 --> 00:00:39,320 experiment. 15 00:00:39,320 --> 00:00:42,910 This is done by specifying a so-called sample space. 16 00:00:42,910 --> 00:00:46,170 And then, we specify a probability law, which assigns 17 00:00:46,170 --> 00:00:50,060 probabilities to outcomes or to collections of outcomes. 18 00:00:50,060 --> 00:00:53,360 The probability law tells us, for example, whether one 19 00:00:53,360 --> 00:00:57,110 outcome is much more likely than some other outcome. 20 00:00:57,110 --> 00:01:00,640 Probabilities have to satisfy certain basic properties in 21 00:01:00,640 --> 00:01:02,070 order to be meaningful. 22 00:01:02,070 --> 00:01:05,069 These are the axioms of probability theory. 23 00:01:05,069 --> 00:01:08,370 For example probabilities cannot be negative. 24 00:01:08,370 --> 00:01:11,940 Interestingly, there will be very few axioms, but they are 25 00:01:11,940 --> 00:01:14,890 powerful, and we will see that they have lots of 26 00:01:14,890 --> 00:01:16,220 consequences. 27 00:01:16,220 --> 00:01:19,400 We will see that they imply many other properties that 28 00:01:19,400 --> 00:01:22,480 were not part of the axioms. 29 00:01:22,480 --> 00:01:25,510 We will then go through a couple of very simple examples 30 00:01:25,510 --> 00:01:27,789 involving models with either discrete 31 00:01:27,789 --> 00:01:29,730 or continuous outcomes. 32 00:01:29,730 --> 00:01:32,690 As you will be seeing many times in this class, discrete 33 00:01:32,690 --> 00:01:35,070 models are conceptually much easier. 34 00:01:35,070 --> 00:01:39,160 Continuous models involve some more sophisticated concepts, 35 00:01:39,160 --> 00:01:43,270 and we will point out some of the subtle issues that arise. 36 00:01:43,270 --> 00:01:45,880 And finally, we will talk a little bit about the big 37 00:01:45,880 --> 00:01:49,560 picture, about the role of probability theory, and its 38 00:01:49,560 --> 00:01:51,220 relation with the real world.