While the previous chapter covered probability on events, in this chapter we will switch to talking about random variables and their corresponding distributions. We will cover the most common discrete distributions, define the notion of a joint distribution, and finish with some practical examples of how to reason about the probability that one device will fail before another.

The general setting in statistics is that we observe some data and then try to infer some property of the underlying distribution behind this data. The underlying distribution behind the data is unknown and represented by random variable (r.v.) $X$. This chapter will briefly introduce the general concept of estimators, focusing on estimators for the mean and variance.

This chapter deals with one of the most important aspects of systems modeling, namely the arrival process. When we say “arrival process” we are referring to the sequence of arrivals into the system. The most widely used arrival process model is the Poisson process. This chapter defines the Poisson process and highlights its properties. Before we dive into the Poisson process, it will be helpful to review the Exponential distribution, which is closely related to the Poisson process.

This chapter begins our study of Markov chains, specifically discrete-time Markov chains. In this chapter and the next, we limit our discussion to Markov chains with a finite number of states. Our focus in this chapter will be on understanding how to obtain the limiting distribution for a Markov chain.

In the last two chapters we studied many tail bounds, including those from Markov, Chebyshev, Chernoff and Hoeffding. We also studied a tail approximation based on the Central Limit Theorem (CLT). In this chapter we will apply these bounds and approximations to an important problem in computer science: the design of hashing algorithms. In fact, hashing is closely related to the balls-and-bins problem that we recently studied in Chapter 19.

This part of the book is devoted to randomized algorithms. A randomized algorithm is simply an algorithm that uses a source of random bits, allowing it to make random moves. Randomized algorithms are extremely popular in computer science because (1) they are highly efficient (have low runtimes) on every input, and (2) they are often quite simple.

In the previous chapter, we studied individual continuous random variables. We now move on to discussing multiple random variables, which may or may not be independent of each other. Just as in Chapter 3 we used a joint probability mass function (p.m.f.), we now introduce the continuous counterpart, the joint probability density function (joint p.d.f.). We will use the joint p.d.f. to answer questions about the expected value of one random variable, given some information about the other random variable.

This final part of the book is devoted to the topic of Markov chains. Markov chains are an extremely powerful tool used to model problems in computer science, statistics, physics, biology, and business – you name it! They are used extensively in AI/machine learning, computer science theory, and in all areas of computer system modeling (analysis of networking protocols, memory management protocols, server performance, capacity provisioning, disk protocols, etc.). Markov chains are also very common in operations research, including supply chain, call center, and inventory management.

We have studied several common continuous distributions: the Uniform, the Exponential, and the Normal. However, if we turn to computer science quantities, such as file sizes, job CPU requirements, IP flow times, and so on, we find that none of these are well represented by the continuous distributions that we’ve studied so far. To understand the type of distributions that come up in computer science, it’s useful to start with a story.

This chapter introduces randomized algorithms. We start with a discussion of the differences between randomized algorithms and deterministic algorithms. We then introduce the two primary types of randomized algorithms: Las Vegas algorithms and Monte Carlo algorithms. This chapter and its exercises will contain many examples of randomized algorithms, all of the Las Vegas variety. In Chapter 22 we will turn to examples of the Monte Carlo variety.