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In this segment, we derive and discuss the weak law of large numbers. It is a rather simple result, but plays a central role within probability theory. The setting is as follows. We start with some probability distribution that has a certain mean and variance, which we assume to be finite. We then draw independent random variables out of this distribution so that these Xi's are independent and identically distributed, i.i.d. for short. What's going on here is that we're carrying out a long experiment during which all of these random variables are drawn. Once we have drawn all of these random variables, we can calculate the average of the values that have been obtained, and this gives us the so-called sample mean. Notice that the sample mean is a random variable because it is a function of random variables. It should be distinguished from the true mean, mu, which is the expected value of the Xi's, which is a number. It is not random. And mu is some kind of ...

In this segment, we derive and discuss the Markov inequality, a rather simple but quite useful and powerful fact about probability distributions. The basic idea behind the Markov inequality as well as many other inequalities and bounds in probability theory is the following. We may be interested in saying something about the probability of an extreme event. By extreme event, we mean that some random variable takes a very large value. If we can calculate that probability exactly, then, of course, everything is fine. But suppose that we only have a little bit of information about the probability distribution at hand. For example, suppose that we only know the expected value associated with that distribution. Can we say something? Well, here's a statement, which is quite intuitive. If you have a non-negative random variable, and I tell you that the average or the expected value is rather small, then there should be only a very small probability that the random variable t...