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In probability theory, the expectation or expected value of a random variable provides a measure of the center or average of the possible outcomes, weighted by their associated probabilities. If you have a random variable, let’s say X, the expected value of X, denoted as E[X], is essentially an average obtained by multiplying each possible outcome by its probability and summing all these products.

For a discrete random variable, the expectation is calculated by summing the product of each outcome and its probability. For a continuous random variable, we use integration to compute the expected value. The formula for the expected value of a continuous random variable X, when dealing with a probability density function f(x), is given by:
\[ E[X] = \bigint_{-\infty}^{\infty} xf(x)dx \]
This concept is pivotal in understanding how to solve the given textbook exercise, as it sets the foundation for evaluating the integrated product of two random variables, X and Y.

A joint probability density function (pdf) characterizes the probability that two continuous random variables will take specific values simultaneously. When working with two random variables, X and Y, their joint probability density function, f(x, y), helps determine the likelihood of X being near x and Y being near y at the same time.

The joint pdf must satisfy two main conditions: it must be non-negative, and the integral of the function over all possible values of x and y must equal one:
\[ \bigint_{-\infty}^{\infty} \bigint_{-\infty}^{\infty} f(x, y)dxdy = 1 \].
Understanding the joint pdf is critical for solving the exercise because it allows us to compute the expectation of the product XY by integrating it with the joint pdf, as the exercise solution illustrates.

Probability theory is the branch of mathematics dealing with the analysis of random phenomena. It provides tools for predicting the outcomes of complex systems under uncertainty. Central to this field are concepts like random variables, probability distributions, expected values, and variances—each serving as building blocks to quantify the likelihood of events and their theoretical assumptions.

In the context of the given exercise, probability theory lays the foundation for understanding how the inequality involving expectations is derived. By acknowledging that Y is less than or equal to a constant c, probability theory allows us to reason about the product XY and its behavior within the given condition. This logical framework helps in applying the relevant properties and operations, such as integration and setting inequality bounds, which are crucial to arriving at the expected value inequality presented in the solution.

In the realm of continuous random variables, integration plays a key role in calculating probabilities and expected values. Integration in probability involves integrating a variable against its probability density function, which requires solid calculus skills. For example, to get the expected value of a random variable over its entire range of values, we integrate the product of the variable and its probability density function over all possible values.

The exercise you are studying showcases the practical application of integration in probability. By setting up integral expressions to represent the expected value, we can accommodate the entire spectrum of potential outcomes weighted by their probabilities. The step-by-step solution displays the integration process, ensuring the conditions (such as the constant bound c on the variable Y) are appropriately applied to achieve the inequality that we desire to show: \( E[XY] \leq cE[X] \).
Understanding integration within the context of probability is therefore essential for finding solutions to such exercises, as it equips you with the methodology to establish the relationships between variables and their expectations under various constraints.