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ALBERT MEYER: In the last lecture, we spent time talking about the mean, or expectation, and its properties, most important one being linearity. But let's step back now and think about, what is it that the mean means? Why we care about it? We have this intuitive idea that if you do things long enough, if you keep experimenting with the same random variable collecting its values, its long run average will be about the same as its mean. Now, we're going to try to make that more precise. So we're going to talk about the topic of deviation from the mean, or as I like to say, what does the mean really mean? Why do we care about it? Well, let's look at an example that's familiar to get a grip on the specific ideas that we're interested in. So suppose I toss a fair coin 101 times. Then, I know that the expected number, since all the values from zero through 101 are possible, and the middle value is the expectation, it's 50 and 1/2 heads. Now, I'm ...

ALBERT MEYER: Today's topic is random variables. Random variables are an absolutely fundamental concept in probability theory. But before we get into officially defining them, let's start off with an example that in fact, is a game because that's a fun way to start. So we're going to play the bigger number game and here's how it works. There are two teams, and Team 1 has the task of picking two different integers between 0 and 7 inclusive, and they write one integer on one piece of paper and the other integer on the other piece of paper. They turn the two pieces of paper face down so the numbers are not visible, and the other team then sees these two pieces of paper whose other side has different numbers written on them sitting on the table. What Team 2 then does is picks one of the pieces of paper and turns it over and looks at the number on it. And then, based on what that number is, they make a decision, stick with the number they have or switch to ...

ALBERT MEYER: We've looked at independence for two events. What about when we have a bunch of events? Well, in that case we want to look at the idea of mutual independence. So let's check that out. I'll say that if I have n different events, I'll say that they're mutually independent, intuitively. If the probability that one of them occurs is unchanged by which other ones happen to have occurred. So expressed in conditional probability, which is the way to make it precise, what we're really saying is that events A1 through An are mutually independent when the probability of Ai is equal to the probability of Ai, given the intersection of any of the other As as long as i is not one of them. So take A1, A2, or A1, A2, A3, and so on. And A5 is going to be independent of all of those other intersections. If we shift over to the other definition of independence that we used for two sets, in terms of products, you could say that n sets are mutually indepe...

ALBERT MEYER: We've considered arithmetic sums, where each term is a fixed amount larger than the previous term by an additive amount. Another kind of sum called geometric sums which, we're going to look at now, come about where each sum is a fixed multiple of the previous sum. And these come up all the time in lots of different settings. And in this particular case, we're going to look at an example of how it applies to analyzing the value of money in the future. So let's begin with a geometric sum. And there's the standard form of it. Geometric sum is of the form 1 plus x plus x squared up through the nth power of x. And, for uniformity, notice that 1 is actually x to the 0. So we're taking the sum from k equals 0 to n of x to the k. Now, what I'd like to do is find a nice closed form for this without those ellipses, and having a growing number of terms n. And there's a simple trick with the arithmetic sum. The way Gauss got it and the wa...

ALBERT MEYER: The topic for this video lecture is Connected Vertices, the kind of connections we can have between different vertices through edges. We're going to start by showing that the shortest walk between two vertices is a path. We're going to prove it by contradiction. Suppose we have some path from u to v and it crosses over itself. So here we have u and v. At some point, you get to c, and you go back to c, and from there you go to v. So you've gone through some vertice twice. But if you want to get the shortest path from u to v, why would you go through this loop? Why not just keep going straight from c to v? The path without that section that goes from c back to itself also for u to v and is shorter. So if we have any path that crosses over itself, we can just get rid of that part that loops around back into itself, and we still have a walk from u to v. So therefore the shortest walk from u to v is going to be a path. Now we're going to talk abou...