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PROFESSOR: Let me discuss, for a second, effective potential. And-- actually, let me ask you a question to get you started. We're going to look at high n. N-- so n's, say, 100. And then l will go from 0 up to 99. Very high l's. What's going to be different from them? Well here is one possibility, and that's a possibility that happens. The orbits are going to go from being circular to being elliptical as you change l. And I ask you-- which l would correspond to circular orbit, and which l would correspond to the most elliptical orbit? So let's take a poll/ so most circular orbit-- is it l equals 0 or l equals 99? Who votes for l equals 0 for the most circular orbit? I have about half [INAUDIBLE], a little bit more than half. OK. Who votes for l equals 99? [INAUDIBLE] Which is the most [INAUDIBLE]? OK, this is funny, because we all associate most circular with l equals 0, spherically symmetric. But it's wrong. This is the most circular. AUDIENCE:...

In this segment, we discuss the so-called "random incidence" paradox for the Poisson process. It's a paradox because it involves a somewhat counterintuitive phenomenon. However, we will understand exactly what's going on, and in the end, it will cease to be a paradox and we will have an intuitive understanding of what exactly is happening. So consider a Poisson process that has been running forever, or think of it as a Poisson process that started a very long time back in the past. To make things concrete, suppose that the arrival rate is 4 arrivals per hour so that the expected interarrival time is one fourth, in hours, or that would be the same as 15 minutes. For example, suppose that the bus company in your town claims that buses arrive to your stop according to a Poisson process with this particular rate. But you don't really believe that your bus company is telling the truth and you decide to investigate. So what you do is the following. You sho...

In this segment, we discuss a little more the mean squared error. Consider some estimator. It can be any estimator, not just the sample mean. We can decompose the mean squared error as a sum of two terms. Where does this formula come from? Well, we know that for any random variable Z, this formula is valid. And if we let Z be equal to the difference between the estimator and the value that we're trying to estimate, then we obtain this formula here. The expected value of our random variable Z squared is equal to the variance of that random variable plus the square of its mean. Let us now rewrite these two terms in a more suggestive way. We first notice that theta is a constant. When you add or subtract the constant from a random variable, the variance does not change. So this term is the same as the variance of theta hat. This quantity here, we will call it the bias of the estimator. It tells us whether theta hat is systematically above or below than the unknown parame...

Let us now discuss in some more detail what it takes to carry out Bayesian inference, when both random variables are discrete. The unknown parameter, Theta, is a random variable that takes values in the discrete set. And we can think of these values as alternative hypotheses. In this case, we know how to do inference. We have in our hands the Bayes rule and we have seen plenty of examples. So instead of going through one more example in detail, let us assume that we have a model, that we have observed the value of X, and that we have already determined the conditional PMF of the random variable Theta. As a concrete example, suppose that Theta can take values 1, 2, or 3. We have obtained our observation, and the conditional PMF takes this form. We could stop at this point or we could continue by asking for a specific estimate of Theta-- our best guess as to what Theta is. One way of coming up with an estimate is to use the maximum a posteriori of probability rule, which lo...

In this segment, we discuss the expected value rule for calculating the expected value of a function of a random variable. It corresponds to a nice formula that we will see shortly, but it also involves a much more general idea that we will encounter many times in this course, in somewhat different forms. Here's what it is all about. We start with a certain random variable that has a known PMF. However, we're ultimately interested in another random variable Y, which is defined as a function of the original random variable. We're interested in calculating the expected value of this new random variable, Y. How should we do it? We will illustrate the ideas involved through a simple numerical example. In this example, we have a random variable, X, that takes values 2, 3, 4, or 5, according to some given probabilities. We are also given a function that maps x-values into y-values. And this function, g, then defines a new random variable. So if the outcome of the ex...

Finally, let's discuss the third class of instructions that let us change the program counter. Up until now, the program counter has simply been incremented by 4 at the end of each instruction, so that the next instruction comes from the memory location that immediately follows the location that held the current instruction, i.e., the Beta has been executing instructions sequentially from memory. But in many programs, such as in factorial, we need to disrupt sequential execution, either to loop back to repeat some earlier instruction, or to skip over instructions because of some data dependency. We need a way to change the program counter based on data values generated by the program's execution. In the factorial example, as long as b is not equal to 0, we need to keep executing the instructions that calculate a*b and decrement b. So we need instructions to test the value of b after it's been decremented and if it's non-zero, change the PC to repeat the lo...

Let's discuss the method Ashenfelter used to build his model, linear regression. We'll start with one-variable linear regression, which just uses one independent variable to predict the dependent variable. This figure shows a plot of one of the independent variables, average growing season temperature, and the dependent variable, wine price. The goal of linear regression is to create a predictive line through the data. There are many different lines that could be drawn to predict wine price using average growing season temperature. A simple option would be a flat line at the average price, in this case 7.07. The equation for this line is y equals 7.07. This linear regression model would predict 7.07 regardless of the temperature. But it looks like a better line would have a positive slope, such as this line in blue. The equation for this line is y equals 0.5*(AGST) -1.25. This linear regression model would predict a higher price when the temperature is higher. Let...