Expectation with Examples in Statistics

 

Certainly! In statistics, the concept of expectation (also known as expected value) is a fundamental measure used to quantify the average outcome of a random variable over a large number of trials. It provides a way to summarize the long-term behavior or central tendency of a random process or distribution. Let's delve deeper into this concept with examples and numerical illustrations.

Definition and Notation:

The expectation of a random variable $X$ , denoted as $E(X)$ or $\mathbb{E}[X]$ , is defined as:

$E(X) = \sum_{x} x \cdot P(X = x) \quad \text{for discrete random variables}$ $E(X) = \int_{-\infty}^{\infty} x \cdot f(x) \, dx \quad \text{for continuous random variables}$

where:

• $X$ is the random variable.
• $P(X = x)$ is the probability mass function (PMF) for discrete $X$ .
• $f(x)$ is the probability density function (PDF) for continuous $X$ .

Examples:

Example 1: Fair Six-Sided Die

Consider a fair six-sided die. Let $X$ be the outcome of a single roll.

• Random Variable $X$ : Possible outcomes are $\{1, 2, 3, 4, 5, 6\}$ .
• Probability Mass Function (PMF): $P(X = x) = \frac{1}{6}$ for $x = 1, 2, 3, 4, 5, 6$ .

To find $E(X)$ :

$E(X) = \sum_{x=1}^{6} x \cdot P(X = x) = \sum_{x=1}^{6} x \cdot \frac{1}{6}$ $E(X) = \frac{1}{6} \cdot (1 + 2 + 3 + 4 + 5 + 6) = \frac{21}{6} = 3.5$

So, the expected value of $X$ , the outcome of a single roll of a fair six-sided die, is $3.5$ .

Example 2: Continuous Uniform Distribution

Let $X$ follow a continuous uniform distribution on the interval $[a, b]$ .

• Random Variable $X$ : $X \sim \text{Uniform}(a, b)$ .
• Probability Density Function (PDF): $f(x) = \frac{1}{b-a}$ for $x \in [a, b]$ .

To find $E(X)$ :

$E(X) = \int_{a}^{b} x \cdot \frac{1}{b-a} \, dx$ $E(X) = \frac{1}{b-a} \cdot \left[ \frac{x^2}{2} \right]_{a}^{b}$ $E(X) = \frac{1}{b-a} \cdot \frac{b^2 - a^2}{2}$ $E(X) = \frac{a + b}{2}$

Thus, the expected value of $X$ for a continuous uniform distribution $\text{Uniform}(a, b)$ is $\frac{a + b}{2}$ .

Importance of Expectation:

• Meaning: The expectation represents the average value of $X$ over a large number of trials or observations.
• Utility: It serves as a measure of central tendency and helps in decision-making and risk assessment.
• Applications: Used in finance (expected returns), physics (expected energy levels), and various fields of engineering and sciences.

Conclusion:

Expectation is a powerful concept in statistics, providing a succinct summary of the behavior of random variables. Whether in discrete or continuous contexts, its calculation offers insights into the average outcome of random phenomena, making it a cornerstone of statistical analysis and probability theory.