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 $p_{XY}(x,y)$ over all possible values of $x$ and $y$. Mathematically, this can be expressed as:

$$\mathbb{E}[g(X,Y)]=\sum _x\sum _{y}g(x,y)\cdot p_{XY}(x,y)$$

Here, $p_{XY}(x,y)$ 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:

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

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

$$\begin{array}{rl}\mathbb{E}[g(X,Y)]& =\sum _x\sum _{y}g(x,y)\cdot p_{XY}(x,y)\\ & =(1^2\cdot 2)\cdot 0.1+(2^2\cdot 1)\cdot 0.2+(2^2\cdot 2)\cdot 0.3+(3^2\cdot 3)\cdot 0.4\\ & =1.2+0.8+4.8+21.6\\ & =28.4\end{array}$$