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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: In order to learn more about this subject, we must do the wave packets. So this is the place where you really connect this need solution of Schrodinger's equation, the energy eigenstates, to a physical problem. So we'll do our wave packets. So we've been dealing with packets for a while, so I think it's not going to be that difficult. We've also been talking about stationary phase and you've practiced that, so you have the math ready. We should not have a great difficulty. So let's new wave packets with-- I'm going to use A equals 1 in the solution. Now, I've erased every solution, and we'll work with E greater than v naught to begin with. The reason I want to work with E greater than v naught is because there is a transmitted wave. So that's kind of nice. So what am I going to do? I'm going to write it this following way. So here is a solution with A equals to 1. e to the i kx plus k minus k bar over k plus k bar...

In this video for 3D geometry, we will be talking about two topics. One is the angle bisector of two planes and one is line of intersection of two planes. Both of the topics are important especially the line of intersection of two planes, and I will be spending some time on that. Angle bisector of two planes is a very easy topic, something you should remember while preparing for JEE. I will just give you the formula and I won’t be doing a problem but I will spending sometime on this (Line of Intersection of two planes), because this is a relatively important topic. So let us say we have a pair of planes. And you have to find the angle bisector. This is the angle bisector — plane which is the angle bisector of two planes. This angle is the same as this angle. You have been given equations for P1 and P2. You have find equation of angle bisector. If you think about this — easiest way to think about planes is using a notebook. I always recommend you do that if you are a...

Hi everyone In this video, we will be talking about two topics of 3D chapter for mathematics of JEE preparation. The topics are: doing problems on finding perpendicular distance of a point from line and we will be finding equations of a line perpendicular to two lines given. These are small topics but as in other maths topics, you should learn the approach to solve problems like these. Let us start by finding perpendicular distance of a point from a line. We have a line given. Let us saw we have point A, and that is a_vector, and we have been given b_vector, which is the parallel vector. And we have been given a point P. Let us call it p_vector. And we have been asked to find the perpendicular distance of the point from the line. You may recall these kind of problems in 2D in straight lines. But now its 3D or 3-Dimensinal Geometry. So how will we solve this problem? One of the approaches to start this to think that there is a point C here, let us call it c_vector. And the...

GUOLONG: That is Friday in one of the classrooms of Building 26, 11:00 AM. MARZIEH: So I was quite nervous. SHAWN: The first day is a little bit nervous for me. CARLA: I was nervous. I'm not going to deny it. NARRATOR: Facing your students for the first time can be a little scary, but it's also an opportunity to establish that your class will be user-friendly. BILLY: I think the first day of class really sets the tone for the rest of the semester. NARRATOR: How can you set the right tone? As they make their way to your classroom on the first day, what qualities are your students hoping for? BILLY: Being knowledgeable and understanding the concept. ADRIAN: And you need to feel that you can approach them. CHANDLER: Have just an air of confidence. ADRIAN: Being easy to understand. BILLY: And being enthusiastic. CHANDLER: Being there, having a presence, having a lesson plan ready. CAROL: They're well-organized. NARRATOR: That may sound like a lot. Let's start ...

In this case study, we're going to see how we can create a photo illustration of your work. The idea here is to show how a liquid battery works for a final illustration. So first, create photographic bits and pieces and put them all together. Think of it just as if you were drawing the pieces and putting those sketches together. After I met with the researchers for a brainstorming meeting, I got all this material from them, including a diagram that described my idea to create a model of how a liquid battery works. This is the way I imagined the shoot to go. OK first, I would take an optical quality cuvette and would then pour the mercury and the water into the cuvette and somehow suspending with a clamp, a strip of metal foam, which is an important part of the science. And because I was imagining a full container of this material after making the image, I would then crop out the right side-- we don't need to see the clamping device-- and then flip the left side, o...

PROFESSOR: Welcome back. So in this session, we're going to look at unit step and impulse responses. So in this question, we ask you to find the unit impulse response to these two equations, x dot plus 2x equals f of t, and 2 x dot dot plus 27 x dot-- oops, it should be a 7-- plus 7 x dot plus 3x equals f of t. In the second part, you're asked to find the unit step response for the first equation. So here, the key points are really to remember what do we mean by unit impulse and unit step response. Which initial condition correspond to these responses? And what functions of f of t do you choose in each case? So why don't you pause the video and work through this problem? And I'll be right back. Welcome back. So let's look at the equation A. So the unit impulse response is simply-- I'm going to write this down, unit impulse response-- is simply the solution to the following problem, to our differential equation, x dot plus 2x that we're given, w...

In December 1998, the Mars climate orbiter was launched in hopes of providing detailed information about the atmosphere of Mars. The launch went well. However the two teams that collaborated on the project worked in different units--one used the metric system, and the other the English system, and the two systems of units were never properly reconciled. So while the orbiter was intended to fall into orbit around Mars, instead it crash landed on the surface. So units are pretty important. This video is part of the Problem Solving video series. Problem-solving skills, in combination with an understanding of the natural and human-made world, are critical to the design and optimization of systems and processes. Hi my name is Ken Kamrin, and I am a mechanical engineering professor at MIT. Today I want to talk to you about using unit analysis in problem solving. This might seem simple, but it is a critical tool for validating your calculations. So pay attention. To understand t...

Hi. In this problem, we're going to get a bunch of practice working with multiple random variables together. And so we'll look at joint PDFs, marginal PDFs, conditional PDFs, and also get some practice calculating expectations as well. So the problem gives us a pair of random variables-- x and y. And we're told that the joint distribution is uniformly distributed on this triangle here, with the vertices being 0, 0 1, 0, and 0, 1. So it's uniform in this triangle. And the first part of the problem is just to figure out what exactly is disjoint PDF of the two random variables. So in this case, it's pretty easy to calculate, because we have a uniform distribution. And remember, when you have a uniform distribution, you can just imagine it being a sort of plateau coming out of the board. And it's flat. And so the height of the plateau, in order to calculate it, you just need to figure out what the area of this thing is, of this triangle is. So rememb...

In this problem, we will be helping Romeo and Juliet meet up for a date. And in the process, also we'll review some concepts in basic probability theory, including sample spaces and probability laws. This problem, the basic setup is that Romeo and Juliet are trying to meet up for a date. And let's say they're trying to meet up for lunch tomorrow at noon. But they're not necessarily punctual. So they may arrive on time with a delay of 0, or they may actually be up to 1 hour late and arrive at 1:00 PM. So the other thing that we assume in this problem is that all pairs of arrival times-- so the time that Romeo arrives paired with the time they Juliet arrives-- all of these pairs are equally likely. And I've put this in quotes, because we haven't really specify exactly what this means. And we'll come back to that in a little bit. The last important thing is that each person will wait for 15 minutes for the other person to arrive. If within that ...

PROFESSOR: This definition in which the uncertainty of the permission operator Q in the state psi. It's always important to have a state associated with measuring the uncertainty. Because the uncertainty will be different in different states. So the state should always be there. Sometimes we write it, sometimes we get a little tired of writing it and we don't write it. But it's always implicit. So here it is. From the analogous discussion of random variables, we were led to this definition, in which we would have the expectation value of the square of the operator minus the square of the expectation value. This was always-- well, this is always a positive quantity. Because, as claim 1 goes, it can be rewritten as the expectation value of the square of the difference between the operator and its expectation value. This may seem a little strange. You're subtracting from an operator a number, but we know that numbers can be thought as operators as well. Opera...

Three-dimensional case. Now, in the future homework, you will be doing the equivalent of this calculation here with the Laplacians-- it's not complicated-- so that you will derive with the current is. And the current must be a very similar formula as this one. And indeed, I'll just write it here. The current is h bar over m, the imaginary part of psi star. And instead of ddx, you expect the gradient of psi. That is the current for the probability in three dimensions. And the analog of this equation, d rho dt plus dj dx equals 0, is d rho dt plus divergence of j is equal to 0. That is current conservation. Perhaps you do remember that from your study of electromagnetism. That's how Maxwell discovered the displacement current when he tried to figure out how everything was compatible with current conservation. Anyway, that argument I'll do in a second so that it will become clearer. So one last thing here-- it's something also-- you can check the units he...