5.1.2 Two Continuous Random Variables

The probability that the observed value of a continuous rv $X$ lies in a one-dimensional set $A$ (such as an interval) is obtained by integrating the pdf $f\left(x\right)$ over the set $A$. Similarly, the probability that the pair $\left(X,Y\right)$ of continuous rv’s falls in a two-dimensional set $A$ (such as a rectangle) is obtained by integrating a function called the joint density function.

joint probability density function

Let $X$ and $Y$ be continuous rv’s. A joint probability density function $f\left(x,y\right)$ for these two variables is a function satisfying $f\left(x,y\right)\ge 0$ and ${\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }f\left(x,y\right)dxdy=1$. Then for any two-dimensional set $A$

$$P\left[\left(X,Y\right)\in A\right]={\iint }_{A}f\left(x,y\right)dxdy$$

In particular, if $A$ is the two-dimensional rectangle $\{\left(x,y\right):a\le x\le b,c\le y\le d\}$, then

$$P\left[\left(X,Y\right)\in A\right]=P\left(a\le X\le b,c\le Y\le d\right)={\int }_a^b{\int }_c^{d}f\left(x,y\right)dydx$$

We can visualize $f\left(x,y\right)$ as specifying a surface at height $f\left(x,y\right)$ above the point $\left(x,y\right)$ in a three-dimensional coordinate system. Then $P\left[\left(X,Y\right)\in A\right]$ is the volume underneath this surface and above the region $A$, analogous to the area under a curve in the case of a single rv. This is illustrated in Figure 5.1.

Figure 5.1 $P\left[\left(X,Y\right)\in A\right]=$ volume under density surface above $A$

EXAMPLE 5.3

The marginal pdf of each variable can be obtained in a manner analogous to what we did in the case of two discrete variables. The marginal pdf of $X$ at the value $x$ results from holding $x$ fixed in the pair $\left(x,y\right)$ and integrating the joint pdf over $y$. Integrating the joint pdf with respect to $x$ gives the marginal pdf of $Y$.

marginal probability density functions

The marginal probability density functions of $X$ and $Y$, denoted by $f_X\left(x\right)$ and $f_Y\left(y\right)$, respectively, are given by

$$f_X\left(x\right)={\int }_{-\infty }^{\infty }f\left(x,y\right)dy\text{ for }-\infty <x<\infty$$ $$f_Y\left(y\right)={\int }_{-\infty }^{\infty }f\left(x,y\right)dx\text{ for }-\infty <y<\infty$$

EXAMPLE 5.4 (EX 5.3 continued)

In Example 5.3, the region of positive joint density was a rectangle, which made computation of the marginal pdf’s relatively easy. Consider now an example in which the region of positive density is more complicated.

EXAMPLE 5.5