#### Overview of this book

Python, one of the world's most popular programming languages, has a number of powerful packages to help you tackle complex mathematical problems in a simple and efficient way. These core capabilities help programmers pave the way for building exciting applications in various domains, such as machine learning and data science, using knowledge in the computational mathematics domain. The book teaches you how to solve problems faced in a wide variety of mathematical fields, including calculus, probability, statistics and data science, graph theory, optimization, and geometry. You'll start by developing core skills and learning about packages covered in Python’s scientific stack, including NumPy, SciPy, and Matplotlib. As you advance, you'll get to grips with more advanced topics of calculus, probability, and networks (graph theory). After you gain a solid understanding of these topics, you'll discover Python's applications in data science and statistics, forecasting, geometry, and optimization. The final chapters will take you through a collection of miscellaneous problems, including working with specific data formats and accelerating code. By the end of this book, you'll have an arsenal of practical coding solutions that can be used and modified to solve a wide range of practical problems in computational mathematics and data science.
Preface
Basic Packages, Functions, and Concepts
Free Chapter
Mathematical Plotting with Matplotlib
Working with Randomness and Probability
Geometric Problems
Finding Optimal Solutions
Miscellaneous Topics
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# Working with random processes

Random processes exist everywhere. Roughly speaking, a random process is a system of related random variables, usually indexed with respect to time t ≥ 0, for a continuous random process, or by natural numbers n = 1, 2, …, for a discrete random process. Many (discrete) random processes satisfy the Markov property, which makes them a Markov chain. The Markov property is the statement that the process is memoryless, in that only the current value is important for the probabilities of the next value.

In this recipe, we will examine a simple example of a random process that models the number of bus arrivals at a stop over time. This process is called a Poisson process. A Poisson process N(t) has a single parameter, λ, which is usually called the intensity or rate, and the probability that N(t) takes the value n at a given time t is given by the following formula:

This equation describes...