12 - TRANSFORMING BIVARIATE DENSITY FUNCTIONS Having seen how to transform the probability density functions associated with a single random variable, the next logical step is to see how to transform bivariate probability density functions. Integration with two Independent Variables Consider f(x1, x2), a function of two independent variables. Using cartesian coordinates, f(x1, x2) might be represented by a surface above the x1-x2 plane; the value of f (x1, x2) at any point (x1, x2) corresponds to the height above the point. With two independent variables, integration is normally expressed with a double-integral sign and integration is over some range R, a specified area in the x1-12 plane: JJ f(x1, 2) dx1 dx2 IR Such an integral represents a volume. If R is a circle then this integral corresponds to the volume of a cylinder standing on R whose upper end is cut by the surface f(x1, x2). The following integral gives the volume of a cone whose height is h and whose base is a circle of radius a centred on the origin (0, 0), this circle being the region R: 11 h(1. a V2 + x:3 ! dx1 dx2 Note that at the centre of the circle val + x2 = 0 and the value of the integrand is h. At the edge of the circle Vx2 + x2 = a and the value of the integrand is 0. In principle, two integrations are carried out in turn: h 1 dx1 dx2 4 2 Vx} + x3 a 2 ! # (12.1) 0 0 In this rearrangement, integration is over one quadrant of the circle and the result is multiplied by 4. For a given value of x2 integration is along a strip one end of which is at x1 = 0 and the other end of which is at x1 = va2 - x2. This accounts for the limits on the inner integration. Already, a seemingly simple example of integration with two independent variables is beginning to become uncomfortably hard! There are better ways of determining the volume of a cone but by judicious substitution of both independent variables even the present approach can be greatly simplified. Integration by Substitution of two new Variables The general formula for integration by substitution of a new variable was given as (11.1): 1. a dr = [id (3) dx dy dy Jy(a) The transformation function is y(x) and its inverse is x(y). - 12.1 -
The equivalent formula when there are two independent variables is: Rx f(x1, 12) dx1 dx2 11. f (x1 (y1, y2), x2(y1, y2) @(y1, y2) a(x1, X2) dy1 dy2 (12.2) Ry There are two transformation functions, y1(x1, x2) and y2(x1, @2), and their inverses are x1(y1, y2) and x1 (y1, y2). The regions Ra and Ry are identical subject to the first being specified in the x1-12 plane and the second being specified in the y1-y2 plane. The item 8(y1, y2) @(x1, X2) is called a Jacobian and is defined as: @(y1, y2) a(x1, X2) 0x1