Probabilities as volumes

We showed earlier that there is zero probability of a univariate continuous random variable, \(X\), taking any specific value, \(x\),

\[ P(X=x) \;=\; 0 \qquad \text{for all } x \]

For a similar reason, there is zero probability associated with specific values for a pair of two continuous random variables, \(X\) and \(Y\),

\[ P(X=x \textbf{ and } Y=y) \;=\; 0 \qquad \text{for all } x,y \]

Events of interest correspond to ranges of values of the variables, such as

\[ P(1 \lt X \lt 2 \textbf{ and } Y \gt 4) \]

For univariate continuous distributions, probabilities are found as areas under a probability density function,

Probabilities for bivariate continuous distributions are defined in an analogous way, but as volumes under a surface in three dimensions. In the illustration below, the surface has unit height for all values of \(x\) and \(y\) between 0 and 1 (and zero height elsewhere). The shaded volume is the probability that \((X+Y)\) has a value greater than 1 and can be found by simple geometry to be ½.

Joint probability density function

The shape of the surface defines the joint distribution of the two variables. This surface is called the variables' joint probability density function. It is often denoted by \(f(x,y)\) and defined by a mathematical formula.

In order to be the joint probability density function of two continuous random variables, a function \(f(x, y)\) must satisfy two properties.

Properties of probability functions

\[ f(x,y) \ge 0 \text{ for all } x, y \] \[ \iint\limits_{\text{all } x,y} f(x,y)\;dx\;dy = 1 \]

The second requirement corresponds to the total volume under the joint probability density function being one. This is required because

\[ P(-\infty \lt X \lt \infty \textbf{ and } -\infty \lt Y \lt \infty) \]

must be one since \(X\) and \(Y\) are certain to have values within these ranges.