Bivariate Distributions

Continuous Bivariate Distributions

Let (X, Y ) denote two random variables defined on a domain of support Λ ⊂ I​R², where we assume Λ is an open set in I​R². Then a function f : Λ → I​R ₊ is a joint bivariate pdf (probability density function) if it has the following properties:

$$\begin{array}{rcl} & & f(x,y) > 0\mathrm{,for}(x,y) \in \Lambda \\ & & \int \int \limits_{\Lambda }f(x,y)dxdy = 1 \\ & & P((X,Y ) \in S) = \int \int \limits_{S}f(x,y)dxdy, \mathrm{for \, any}S \subset \Lambda \\ \end{array}$$

(1)

The joint cdf (cumulative distribution function) is given by:

$$F(x,y) = P(X \leq x,Y \leq y) =\int_{-\infty }^{y}\int_{-\infty }^{x}f(v,w)dvdw$$

(2)

where 0 ≤ F(x, y) ≤ 1. The probability content of a rectangular region S = { (x, y) : a < x < b...