Introduction

In the study of multivariate random variables, understanding conditional expectations is crucial. Conditional expectations provide a way to predict the value of one random variable given the value of another, thus offering insights into the relationship between two variables. In this chapter, we will delve into the concept of computing the conditional expectation of a component of a bivariate random variable. We will explore how this computation is carried out and examine its significance in probability and statistics.

Computing the Conditional Expectation

Conditional expectation is the expected value of one random variable given the value of another. In the context of a bivariate random variable (X, $Y$ ), the conditional expectation of a component, say $X$, given the value of the other component $Y=y$, is denoted as $E(X \mid Y=y)$.

The formula for computing the conditional expectation of $\mathrm{X}$ given $\mathrm{Y}=\mathrm{y}$ is given by:

$$E(X \mid Y=y)=\sum_x x \cdot P(X=x \mid Y=y)$$

This formula involves summing the product of the possible values of $X$ (denoted as $\mathrm{X}$ ) and their corresponding conditional probabilities, given that $Y=y$.

Example: Consider a bivariate random variable $(X, Y)$ representing the age $(X)$ and income $(Y)$ of individuals. To compute the conditional expectation of age $(X)$ given that income $(Y)$ is 50,000$ USD, we follow these steps:

• Identify the values of $X$ that correspond to Y= 50,000 USD.
• Calculate the conditional probabilities $P(X=x \mid Y= 0,000)$ for each value of $X$.
• Compute the product of each value of $X$ and its conditional probability, then sum up these products.

Significance

The conditional expectation of a component of a bivariate random variable provides insights into the relationship between the variables. It serves as a predictive measure, indicating the average value of X when Y takes on a specific value. This concept is particularly useful in decision-making, risk assessment, and understanding the conditional behavior of variables.