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An interesting random variable associated with the Bernoulli process is the time of the kth success or the time of the kth arrival, depending on what kind of context we have in mind. So the picture is as follows. The process starts and we wait until the first arrival occurs, and the time that it occurs, we call that time Y1. Then we keep observing the process, and there's a time at which a second arrival comes. We call that time Y2. The process continues, and there is a certain time that the third arrival comes. We call that time Y3. Now, the time that the first arrival comes, this is also what we called T1. T1 is this length. It's the time until the first arrival. Let us give a name to the time it takes from the first to the second arrival, and we call that time T2, which is the second inter-arrival time. And similarly, we will call T3 the time between the second and the third arrival. So we define in general Tk to be the difference between two consecutive arriva...

Our discussion of random variable so far has involved nothing but standard probability calculations. Other than using the PMF notation, we have done nothing new. It is now time to introduce a truly new concept that plays a central role in probability theory. This is the concept of the expected value or expectation or mean of a random variable. It is a single number that provides some kind of summary of a random variable by telling us what it is on the average. Let us motivate with an example. You play a game of chance over and over, let us say 1,000 times. Each time that you play, you win an amount of money, which is a random variable, and that random variable takes the value 1, with probability 2/10, the value of 2, with probability 5/10, the value of 4, with probability 3/10. You can plot the PMF of this random variable. It takes values 1, 2, and 4. And the associated probabilities are 2/10, 5/10, and 3/10. How much do you expect to have at the end of the day? Well, if ...

The last discrete random variable that we will discuss is the so-called geometric random variable. It shows up in the context of the following experiment. We have a coin and we toss it infinitely many times and independently. And at each coin toss we have a fixed probability of heads, which is some given number, p. This is a parameter that specifies the experiment. When we say that the infinitely many tosses are independent, what we mean in a mathematical and formal sense is that any finite subset of those tosses are independent of each other. I'm only making this comment because we introduced a definition of independence of finitely many events, but had never defined the notion of independence or infinitely many events. The sample space for this experiment is the set of infinite sequences of heads and tails. So a typical outcome of this experiment might look like this. It's a sequence of heads and tails in some arbitrary order. And of course, it's an infinite...

A random variable can take different numerical values depending on the outcome of the experiment. Some outcomes are more likely than others, and similarly some of the possible numerical values of a random variable will be more likely than others. We restrict ourselves to discrete random variables, and we will describe these relative likelihoods in terms of the so-called probability mass function, or PMF for short, which gives the probability of the different possible numerical values. The PMF is also sometimes called the probability law or the probability distribution of a discrete random variable. Let me illustrate the idea in terms of a simple example. We have a probabilistic experiment with four possible outcomes. We also have a probability law on the sample space. And to keep things simple, we assume that all four outcomes in our sample space are equally likely. We then introduce a random variable that associates a number with each possible outcome as shown in this di...