Expectation of a Discrete Random Variable with Coin Tosses

Let X represent the number of heads when this coin is tossed twice. Here's the probability distribution of X. Suppose we want to calculate the expectation of the random variable X.

In this problem, we explore the concept of expectation within the realm of discrete probability. The expectation, often referred to as the expected value, is a fundamental concept that provides a measure of the center of a probability distribution, or in simpler terms, the average outcome one would anticipate over many trials. When dealing with a random variable like X, which represents the number of heads in two tosses of a coin, understanding how to calculate this expectation is crucial.

The expectation is calculated as the weighted sum of all possible values a random variable can take on, where each value is weighted by its probability of occurring. For students in a discrete mathematics course, mastering this concept involves recognizing the relationship between probability distributions and statistical, or expected, outcomes. Given a random variable and its probability distribution, the calculation of expectation is straightforward yet highly insightful. It not only predicts the average result if an experiment were repeated many times but also serves as a stepping stone to more advanced probabilistic concepts like variance and standard deviation.

In this context, calculating the expectation of X requires identifying all possible outcomes—no heads, one head, or two heads—and their respective probabilities from the distribution. This exercise enhances an understanding of how discrete random variables behave and lays the groundwork for comprehending more complex random processes.