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.

In this part of the book we delve deeply into understanding the tail of a random variable, namely the probability that the random variable exceeds some value. While we briefly touched on this topic in Section 5.9, in Chapter 18 we derive much more sophisticated tail bounds, including Chernoff bounds and Hoeffding bounds.

In Chapter 3, we studied several common discrete distributions. In this chapter we will learn how to obtain their mean, or expectation. We will also cover some useful tools that help us to simplify deriving expectations, such as the linearity of expectation result and deriving expectations by conditioning.

In Chapter 6, we covered a type of generating function known as the z-transform, which is particularly well suited to discrete, integer-valued, random variables. In this chapter, we will introduce a new type of generating function, called the Laplace transform, which is particularly well suited to common continuous random variables.

An important and ubiquitous continuous distribution is the Normal distribution (also called the Gaussian). Normal distributions occur frequently in statistics, economics, natural sciences, and social sciences. For example, IQs approximately follow a Normal distribution. Men’s heights and weights are approximately Normally distributed, as are women’s heights and weights.

We have alluded to the fact that probability is useful in the performance analysis and design of computer systems. Queueing theory is an area of applied probability which directly targets systems performance. Here the “system” might refer to a computer system, a call center, a healthcare system, a manufacturing system, a banking system, or one of many other examples. Markov chains (particularly continuous-time chains) are just one of many tools used in queueing theory. In this final part of the book, we provide a very brief introduction to queueing theory. For a much more in-depth coverage, see [35].

At this point, we have discussed many discrete and continuous distributions. This chapter shows how we can generate instances of these distributions and others. This is helpful when performing simulations of computer systems, as in Chapter 14. For example, we might have a computer system where the interarrival times of jobs are well modeled by an Exponential distribution and the job sizes (service requirements) are well modeled by a Pareto distribution. To simulate the system, we need to be able to generate instances of Exponential and Pareto random variables.

So far we have only talked about finite-state discrete-time Markov chains (DTMCs) with $M$ states. Now we move on to infinite-state DTMCs. For a Markov chain with an infinite number of states, one can still imagine a transition probability matrix, $\mathbf{P}$, but the matrix has infinite dimension.

In this part of the book, we repeat the material in Part II, but this time we focus on continuous random variables, which can take on an uncountable number of values. Continuous random variables are very relevant to computer systems – how else can we model response time, for example? Working in continuous time also allows us to leverage everything we know about calculus.

In this chapter we introduce probability on events. We follow an axiomatic approach that uses elementary set theory.

In Chapter 16, we defined an estimator of some unknown quantity, $\theta$, based on experimentally sampled data, $X$. This estimator, denoted by ${\widehat{\theta }}_{\text{ML}}(X)$, is called a maximum likelihood (ML) estimator, because it returns that value of $\theta$ that produces the highest likelihood of witnessing the particular sampled data.

The goal of this part of the book is to learn how to run simulations of computer systems. Simulations are an important part of evaluating computer system performance. For example, we might have a new load-balancing algorithm, and we’re trying to understand whether it reduces the mean job response time or improves utilization. Or we might have a queueing network, where we want to understand the fraction of packet drops when we double the arrival rate of packets. Being able to simulate the computer system is an easy way to get answers to such questions.

One of the most important problems in the field of computer science is a math problem as well: How can we determine if an integer $n$ is prime?