Stat 5101 Lecture Notes - School of Statistics

1.6. MULTIVARIABLE CHANGE OF VARIABLES 27The joint density of X and Y is f X (x)f Y (y) by independence. By the changeof-variableformula, the joint density of U and V isf U,V (u, v) =f X,Y (u − v, v)|J(u, v)|= f X (u − v)f Y (v)We find the marginal of U by integrating out V∫f U (u) = f X (u−v)f Y (v)dvwhich is the convolution formula.Noninvertible TransformationsWhen a change of variable Y = g(X) is not invertible, things are muchmore complicated, except in one special case, which is covered in this section.Of course, the general change of variable theorem (Theorem 1.1) always applies,but is hard to use.The special case we are interested in is exemplified by the univariate changeof variablesR −→ g[0, ∞)defined byg(x) =x 2 , x ∈ R 2 . (1.31)This function is not invertible, because it is not one-to-one, but it has two “sortof” inverses, defined byandh + (y) = √ y, y ≥ 0. (1.32a)h − (y) =− √ y, y ≥ 0. (1.32b)Our first task is to make this notion of a “sort of” inverse mathematicallyprecise, and the second is to use it to get a change of variable theorem. In aid ofthis, let us take a closer look at the notion of inverse functions. Two functions gand h are inverses if, first, they map between the same two sets but in oppositedirectionsS −→ gTS ←− hTand, second, if they “undo” each other’s actions, that is,h[g(x)] = x, x ∈ S (1.33a)andg[h(y)] = y, y ∈ T. (1.33b)