Bi-variate Distributions

Bivariate Distributions

Joint Distribution function: Let X and Y be two random variables defined on the same probability space $(\Omega, \tilde{A}, P[.])$. Then (X, Y) is called a two-dimensional random variable.  The joint cumulative distribution function or joint distribution function of X and Y, denoted by $F_{X,Y}(x,y)$, is defined as

$$F_{X,Y}(x,y)= P[X \leq x, Y \leq y]~ \forall x, y \in R $$

It may be observed that the joint distribution function is a function of two variables and its domain is the xy-plane. Sometimes we write $F_{X,Y}(x,y)$ as $F(x,y)$.

Properties of Joint Distribution function:

1.    If $x_1<x_2$ and $y_1<y_2$ then (rectangle rule)

$$ P[x_1 <X \leq x_2, y_1<Y\leq y_2]=F(x_2,y_2)-F(x_2,y_1)-F(x_1,y_2)+F(x_1, y_1) \geq 0$$

2.    (a) $F(-\infty, y)=\lim_{x \to -\infty} F(x,y)=0 ~\forall y \in R$

       (b) $F(x, -\infty)=\lim_{y \to -\infty}F(x,y)=0 ~\forall x \in R$

       (c) $F(x, -\infty)= \lim_{x\to \infty, y\to \infty}F(x,y)=1$

3. $F(x, y)$ is right continuous in each argument i.e.

$$\lim_{h\to 0_+} F(x+h, y) = lim_{h \to 0_+} F(x, y+h)=F(x, y)$$

Remark: Any function of two variables which fails to satisfy one of the above three conditions is not a joint distribution function.

Example 1. Show that the bivariate function: 

$F(x, y) = \left\{\begin{array}{ll} e^{-(x+y) & \quad x>0, y>0 \\ 0 & \quad otherwise\end{array}\right.$

is not a joint distribution function.

Solution: $x^2$