Random vectors and joint distributions  (Page 2/3)

We formalize as follows:

A pair $\{X,Y\}$ of random variables considered jointly is treated as the pair of coordinate functions for a two-dimensional random vector $W=(X,Y)$ . To each $\omega \in \Omega$ , W assigns the pair of real numbers $(t,u)$ , where $X\left(\omega \right)=t$ and $Y\left(\omega \right)=u$ . If we represent the pair of values $\{t,u\}$ as the point $(t,u)$ on the plane, then $W\left(\omega \right)=(t,u)$ , so that

is a mapping from the basic space Ω to the plane R ² . Since W is a function, all mapping ideas extend. The inverse mapping $W^{-1}$ plays a role analogous to that of the inverse mapping $X^{-1}$ for a real random variable. A two-dimensional vector W is a random vector iff $W^{-1}\left(Q\right)$ is an event for each reasonable set (technically, each Borel set) on the plane.

A fundamental result from measure theory ensures

$W=(X,Y)$ is a random vector iff each of the coordinate functions X and Y is a random variable.

In the selection example above, we model $X$ (the number of juniors selected)   and Y (the number of seniors selected) as random variables. Hence the vector-valued function

Induced distribution and the joint distribution function

In a manner parallel to that for the single-variable case, we obtain a mapping of probability mass from the basic space to the plane. Since $W^{-1}\left(Q\right)$ is an event for each reasonable set Q on the plane, we may assign to Q the probability mass

Because of the preservation of set operations by inverse mappings as in the single-variable case, the mass assignment determines $P_{XY}$ as a probability measure on the subsets of the plane R ² . The argument parallels that for the single-variable case. The result is the probability distribution induced by $W=(X,Y)$ . To determine the probability that the vector-valued function $W=(X,Y)$ takes on a (vector) value in region Q , we simply determine how much induced probability mass is in that region.

As in the single-variable case, we have a distribution function.

Definition

The joint distribution function $F_{XY}$ for $W=(X,Y)$ is given by

This means that $F_{XY}(t,u)$ is equal to the probability mass in the region $Q_{tu}$ on the plane such that the first coordinate is less than or equal to t and the second coordinate is less than or equal to u . Formally, we may write

Now for a given point $(a,b)$ , the region $Q_{ab}$ is the set of points $(t,u)$ on the plane which are on or to the left of the vertical line through $(t,0)$ and on or below the horizontal line through $(0,u)$ (see Figure 1 for specific point $t=a,u=b$ ). We refer to such regions as semiinfinite intervals on the plane.