9.1.2. Expectation for Discrete Random Variables

Let 𝑋 be a discrete random variable taking values 𝑥₁, 𝑥₂,..., 𝑥ₙ with corresponding probabilities $P(X = x_i) = p_i$, where $\sum_{i=1}^{n} p_i = 1$.

This formula describes a discrete random variable, which is a variable that can take on a countable number of distinct values. The variable $X$ is defined such that it can assume values like $x₁, x₂,$ etc. Each value has an associated probability denoted as $p_i$, and the sum of all probabilities for all possible outcomes must equal 1. This indicates that one of the possible outcomes will definitely occur when the experiment is conducted.

Think of rolling a die. The discrete random variable $X$ can take values from 1 to 6, and each outcome has a probability of $\frac{1}{6}$. This scenario illustrates how individual outcomes are associated with specific probabilities.

📌 Formula: $E(X) = \sum_{i=1}^{n} x_i \cdot p_i$

The expectation or mean of the discrete random variable $X$ is calculated by taking the sum of each possible value multiplied by its corresponding probability. Essentially, you are finding a weighted average where each value contributes to the final mean based on how likely it is to occur.

Imagine a game where drawing cards determines your points. If drawing a card has different points (5 points for 'Ace', 3 points for '2', etc.) but you draw cards based on certain probabilities, the expected points give you an understanding of how many points you can anticipate scoring over many games.

✅ Example: Let 𝑋 be the outcome of a fair 6-sided die. Then: $E(X) = \sum_{i=1}^{6} i \cdot \frac{1}{6} = (1 + 2 + 3 + 4 + 5 + 6) \cdot \frac{1}{6} = \frac{21}{6} = 3.5$

In this example, the outcome of rolling a fair die reflects a simple case of calculating expectation. Each of the outcomes (1 through 6) is equally probable with a probability of $\frac{1}{6}$. You sum the products of each outcome and its probability, which gives you 3.5. This is the average value you would expect if you were to roll the die many times.

Consider this like a classroom of students where if each student can score between 1 and 6 on a test, and each score has a similar chance of happening. Over time, if you were to average all the scores of multiple tests, you would find that the average score trends around 3.5, indicating that half the time scores are above this average and half below.

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