Introduction

In this chapter, we will dive into the fundamental concept of the mathematical expectation of a random variable. The mathematical expectation, often referred to as the expected value, is a crucial measure that provides insight into the average outcome of a random variable over a large number of trials or observations. Understanding how to calculate and interpret the mathematical expectation is essential for various applications in probability theory, statistics, and decision-making.

The mathematical expectation of a random variable is a measure that represents the long-term average value of the variable. It provides a way to quantify what we can expect on average when the random variable is repeatedly observed or experimented with. The expectation is a central concept in probability and statistics and is used to make informed decisions based on uncertain outcomes.

Expectation for Discrete Random Variables

For a discrete random variable $X$, the mathematical expectation $E(X)$ is calculated by summing the products of each possible value of $X$ and its corresponding probability. Mathematically, it is expressed as:

$$E(X)=\sum_x x \cdot P(X=x)$$ Where:

• $E(X)$ is the mathematical expectation of $X$
• $x$ represents each possible value of the random variable $X$
• $P(X=x)$ is the probability of $X$ taking the value $x$

Example: Consider rolling a fair six-sided die. Let $X$ represent the outcome of the roll. The possible outcomes are ${1,2,3,4,5,6}$ with equal probabilities. The expectation of $X$ is:

$$E(X)=\frac{1}{6}(1)+\frac{1}{6}(2)+\frac{1}{6}(3)+\frac{1}{6}(4)+\frac{1}{6}(5)+\frac{1}{6}(6)=\frac{21}{6}=3.5$$

Expectation Properties and Applications

• Linearity of Expectation: The expectation of a linear combination of random variables is equal to the linear combination of their individual expectations. Mathematically, for constants $a$ and $b$ and random variables $X$ and $Y$:

$$E(a X+b Y)=a E(X)+b E(Y)$$

• Expectation of a Function of a Random Variable: For a function $g(X)$ of a random variable $X$, the expectation of $g(X)$ is calculated as:

$$E(g(X))=\sum_x g(x) \cdot P(X=x)$$

• Expectation and Decision-Making: The expectation is used in decision theory to make optimal choices based on expected outcomes. It helps in minimizing risks and maximizing gains.