Introduction

In the realm of multivariate random variables, understanding how to compute the expectation of a function for a bivariate discrete random variable is a crucial skill. Expectation, often referred to as the average or mean value of a random variable, takes on a nuanced form when applied to functions of two or more random variables. In this chapter, we will delve into the intricacies of computing the expectation of a function for a bivariate discrete random variable, uncovering the underlying principles and techniques to navigate this aspect of probability theory.

Computation of Expectation for a Bivariate Discrete Random Variable

When dealing with a bivariate discrete random variable, the concept of expectation extends to functions involving both variables. Given a function $g(X, Y)$ of two discrete random variables $X$ and $Y$, the expectation $\mathbb{E}[g(X, Y)]$ is computed by summing the product of the function value and the joint probability mass function $p_{X Y}(x, y)$ over all possible values of $x$ and $y$. Mathematically, this can be expressed as:

$$\mathbb{E}[g(X, Y)]=\sum_x \sum_y g(x, y) \cdot p_{X Y}(x, y)$$

Here, $p_{X Y}(x, y)$ represents the joint probability mass function of the bivariate random variables $X$ and $Y$, and the summation is taken over all possible values of $x$ and $y$ that the variables can take.

Example: Consider two bivariate discrete random variables $X$ and $Y$ with the following joint probability mass function:

$$p_{X Y}(x, y)= \begin{cases}0.1, & \text { for } x=1, y=2 \\ 0.2, & \text { for } x=2, y=1 \\ 0.3, & \text { for } x=2, y=2 \\ 0.4, & \text { for } x=3, y=3 \\ 0, & \text { otherwise }\end{cases}$$

Let’s compute the expectation of the function $g(X, Y)=X^2 \cdot Y$ :

$$\begin{aligned} \mathbb{E}[g(X, Y)] & =\sum_x \sum_y g(x, y) \cdot p_{X Y}(x, y) \\ & =\left(1^2 \cdot 2\right) \cdot 0.1+\left(2^2 \cdot 1\right) \cdot 0.2+\left(2^2 \cdot 2\right) \cdot 0.3+\left(3^2 \cdot 3\right) \cdot 0.4 \\ & =1.2+0.8+4.8+21.6 \\ & =28.4 \end{aligned}$$