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Last time we saw things about gradients and directional derivatives. Before that we studied how to look for minima and maxima of functions of several variables. And today we are going to look again at min/max problems but in a different setting, namely, one for variables that are not independent. And so what we will see is you may have heard of Lagrange multipliers. And this is the one point in the term when I can shine with my French accent and say Lagrange's name properly. OK. What are Lagrange multipliers about? Well, the goal is to minimize or maximize a function of several variables. Let's say, for example, f of x, y, z, but where these variables are no longer independent. They are not independent. That means that there is a relation between them. The relation is maybe some equation of the form g of x, y, z equals some constant. You take the relation between x, y, z, you call that g and that gives you the constraint. And your goal is to minimize f only of tho...

PROFESSOR: We just saw some random variables come up in the bigger number game. And we're going to be talking now about random variables, just formally what they are and their definition of independence for random variables. But let's begin by looking at the informal idea. Again, a random variable is a number that's produced by a random process. So a typical example that comes up where you get a random variable is you've got some system that you're watching and you're going to time it to see when the next crash comes, if it crashes. So assuming that this is unpredictable that it happens in some random way, then the number of hours from the present until the next time the system crashes is a number that's produced by this random process of whether the system works or not. Number of faulty pixels in a monitor. When you're building the monitors and delivering them to the actual computer manufacturers, there's a certain probability that som...

We saw in the previous video that the outcome of a logistic regression model is a probability. Often, we want to make an actual prediction. Should we predict 1 for poor care, or should we predict 0 for good care? We can convert the probabilities to predictions using what's called a threshold value, t. If the probability of poor care is greater than this threshold value, t, we predict poor quality care. But if the probability of poor care is less than the threshold value, t, then we predict good quality care. But what value should we pick for the threshold, t? The threshold value, t, is often selected based on which errors are better. You might be thinking that making no errors is better, which is, of course, true. But it's rare to have a model that predicts perfectly, so you're bound to make some errors. There are two types of errors that a model can make -- ones where you predict 1, or poor care, but the actual outcome is 0, and ones where you predict 0, or g...

So we saw in the last video why if you represent scheduling constraints among courses by a digraph that it's critical that that digraph in fact be a DAG. And let's now look at this scheduling issue represented by DAGs in more detail. So here's a chart of a selection of Course 6 prerequisites-- some of them obsolete, but they serve the purposes of being an illustrative example-- and the little arrows here are indicating arrows in the digraph. So what this tells me is that 18.01 is listed as an immediate prerequisite in the catalog for 6.042. 18.01 is also an immediate prerequisite of 18.02. 6.001 and 6.004 are both prerequisites of 6.033, and 6.042 of 6.046, and 6.046 of 6.840. So we're seeing here this indirect prerequisite issue that I mentioned before, which is that even though the only thing listed as a prerequisite for 6.840 in the catalog is 6.046, as a matter of fact in order to take 6.046 you have to have taken 6.042. So 6.042 is an indirect prerequ...