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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...

GILBERT STRANG: OK. This video is about the third of the great trio of partial differential equations. Laplace's equation was number one. That's called an elliptic equation. The heat equation was number two. That's called a parabolic equation. Now we reach the wave equation. That's number three, and it's called a hyperbolic equation. So somehow the three equations remind us of ellipses, parabolas, and hyperbolas. They have different types of solutions. Laplace's equation, you solve it inside a circle or inside some closed region. The heat equation and the wave equation, time enters, and you're going forward in time. The heat equation is first order in time, du dt. And the wave equation, the full-scale wave equation, is second order in time. That stands for the second derivative, d second u dt squared. And it matches the second derivative in space with a velocity coefficient c squared. I'm in one-dimensional space. If I were in three dimensi...

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...

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...

Take a look at this music box for our first case study. So I place it on the glass of the scanner, and it looks pretty good. But as I look carefully. Which I hope you will do when you try this sort of stuff with your work, a couple of things that I'm not thrilled with. First of all, we're seeing some writing on the barrel. Looking closer, it appears to be numbers and letters. That's not great. Our eyes go straight to letters and numbers, and they basically attract our eye and actually become a distraction from the image, as far as I'm concerned. So I'm not happy with that. I'm also not thrilled with the positioning of the turning mechanism. Let's call it a stem. Somehow it doesn't feel composed properly. So what I simply did, was I played it a bit so the barrel no longer shows the writing, and I turned the stem so that it landed in a better composed image. I've been thinking about composition all my professional life, so see if you can ...

This week we're going to take a look at exactly what you DO with the photographs after you've made them. And, once again, frankly we could devote a full six week course to just this idea alone--presenting your work. But we'll try to cover this topic as best we can in just a couple of tutorials for you for this week. So let's start with a few figures just to dissect them and get you to think more about how you create figures with your images. Let's first take a look at a figure that the researchers made for a grant submission. And here we see a host of the pieces of devices, and the full devices, all labeled. They wanted to show what they were capable of creating--of course which is important. But there's so much going on, the question is whether we want to bother looking anywhere. By the way, just a note, I didn't make any of these images of the research, but I knew that a few of the original images were in color. So first of all, why not use t...

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'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...

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...