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Hi. In this problem, we'll get more practice using conditioning to help us calculate expectations of variances. We'll see that in this problem, which deals with widgets and crates, it's actually similar in flavor to an earlier problem that we did, involving breaking a stick twice. And you'll see that in this problem, we'll again use the law of iterated expectations and the law of total variance to help us calculate expectations of variances. And again, we'll be taking the approach of attacking the problem by splitting into the stages and building up from the bottom up. So in this problem, what we have is a crate, which contains some number of boxes. And we don't know how many boxes are. It's random. And it's given by some discrete random variable, n. And in each box, there are some number of widgets. And again, this is also random. And in each box, say for Box I, there are xi number of widgets in each one. What we're really intere...

PROFESSOR: We'll begin by discussing the wave packets and uncertainty. So it's our first look into this Heisenberg uncertainty relationships. And to begin with, let's focus it as fixed time, t equals zero. So we'll work with packets at t equals zero. And I will write a particular wave function that you may have at t equals 0, and it's a superposition of plane waves. So it would be e to the ikx. You sum over many of them, so you're going to sum over k, but you're going to do it with a weight, and that's 5k. And there's a lot to learn about this, but the physics that is encoded here is that any wave at time equals 0, this psi of x at time equals 0, can be written as a superposition of states with momentum h bar k. You remember e to the ikx represents a particle or a wave that carries momentum h bar k. So this whole idea here of a general wave function being written in this way carries physical meaning for us. It's a quantum mechanical...

This week we'll be talking about probably one of the most important aspects of photography, which is light. As in photo-graphy [spoken phonetically], writing or drawing with light. It's a very content-rich conversation, we're going to have, and could really, easily, take up a full six-week course. But for our purposes here, we're going to give you something of an overview, showing you various forms and characteristics of some light sources. We can't address every conceivable possibility. But what we can do is tweak your curiosity, I hope, so that you pay attention, very close attention, to what you're seeing as you change your own light sources, looking at the shadows for example. How do they and they and the highlights change as you move your light source? And observe exactly what's going on when you make even some very minimal changes. Let's start with a very simple piece of equipment, a lightbox. It's really easy to use when you...

Hi. In this problem, we'll get a chance to see the usefulness of conditioning in helping us to calculate quantities that would otherwise be difficult to calculate. Specifically, we'll be using the law of iterated expectations and the law of total variance. Before we get started, let's just take a quick moment to interpret what these two laws are saying. Really, what it's saying is, in order to calculate the expectation or the variance of some random variable x, if that's difficult to do, we'll instead attack this problem in stages. So the first stage is, we'll condition on some related random variable, y. And the hope is that by conditioning on this and reducing it to this conditional universe, the expectation of x will be easier to calculate. Now, recall that this conditional expectation is really a random variable, which is a function of the random variable y. So what we've done is we first average out x given some y. What remains is so...

Hi, In this problem, we'll be looking at the PDF the absolute value of x. So if we know a random variable, x, and we know it's PDF, how can we use that information to help us find the PDF of another random variable-- the absolute value of x? And so throughout this problem, we'll define a new random variable called y. And we'll define that y to be equal to the absolute value of x, just to make things simpler. So we'll do a couple of concrete examples, and then we'll try to generalize at the end. The first example that we'll deal with in part A is this PDF for x. So we're told that the PDF of x is 1/3 between negative 2 and 1, and 0 otherwise. And here's a picture of what it looks like. It's just a rectangle from negative 2 to 1. So now we want to find out what is the PDF of the absolute value of x, which we've called y? And at this point, it may be helpful to step back and think about this problem from the discrete point of vie...

Hi. In the session, we'll be solving the Monty Hall problem. And this problem is based on an old game show that was called "Let's Make a Deal." And the host of this game show, his name was Monty Hall, which is why this problem is now known as the Monty Hall problem. And this problem is actually pretty well-known, because there was some disagreement at the time over what the right answer to this problem should be. Even some really smart people didn't agree on what the right answer should be. And part of what might explain that disagreement is that they probably were considering slightly different variations of the problem, because as in all probability problems, the assumptions that you're working with are very important, because otherwise you may be solving an actually different problem. And so what we'll do first is really layout concretely what all the assumptions are, what the rules of the game are. And then we'll go through the meth...

PROFESSOR: OK. We'll go through these together. Repeat after me. Civil. AUDIENCE: Civil. PROFESSOR: Civility. AUDIENCE: Civility. PROFESSOR: Civilized. AUDIENCE: Civilized. PROFESSOR: Civilization. . AUDIENCE: Civilization. PROFESSOR: Equal. AUDIENCE: Equal. PROFESSOR: Equality. AUDIENCE: Equality. PROFESSOR: Equalize. AUDIENCE: Equalize. PROFESSOR: Equalization. AUDIENCE: Equalization. PROFESSOR: Fertile. AUDIENCE: Fertile. PROFESSOR: Fertility. AUDIENCE: Fertility. PROFESSOR: Fertilize. AUDIENCE: Fertilize. PROFESSOR: Fertilization. AUDIENCE: Fertilization. PROFESSOR: Final. AUDIENCE: Final. PROFESSOR: Finality. AUDIENCE: Finality. PROFESSOR: Finalize. AUDIENCE: Finalize. PROFESSOR: Finalization. AUDIENCE: Finalization. PROFESSOR: General. AUDIENCE: General. PROFESSOR: Generality. AUDIENCE: Generality. PROFESSOR: Generalized. AUDIENCE: Generalized. PROFESSOR: Generalization. AUDIENCE: Generalization. PROFESSOR: Hospital. AUDIENCE: Hospital. PROFESSOR: Hospitality. A...

GUEST SPEAKER: And what we'll be doing is getting started with our panel discussion. We have structured this discussion to leave a lot of time for questions and answers and discussion from the floor. So this afternoon, we've heard about ways in which OpenCourseWare and similar resources are being transformational in how people get access to information and how they use it. Now I want to turn our attention more to the future and the changes we're likely to see in the years to come. My pleasure to introduce this panel-- the moderator for the panel is professor Hal Abelson. He's a class of 1922 professor of computer science and engineering in the MIT department of electrical engineering and computer science. He's joined in the panel by three distinguished individuals-- Dr. Charles Vest, who, of course, all of us here know extraordinarily well, is currently the president of the National Academy of Engineering and of course the president emeritus of MIT. Jo...