Stat 5101 Lecture Notes - School of Statistics

1.6. MULTIVARIABLE CHANGE OF VARIABLES 25Proof. The general change of variable theorem (Theorem 1.1) saysP Y (A) =P X (B) (1.28)whereB = { x ∈ S : g(x) ∈ A }where S is the sample space of the random vector X, which we may take to bethe open subset of R n on which g is defined. Because g is invertible, we havethe relationship between A and BB = h(A)A = g(B)Rewriting (1.28) using the definition of measures in terms of densities gives∫∫∫f Y (y) dy = f X (x) dx = f X (x) dx (1.29)ABh(A)Now applying Theorem 1.5 to the right hand side gives∫∫f Y (y) dy = f X [h(y)] ·|J(y)|dy.AAThis can be true for all sets A only if the integrands are equal, which is theassertion of the theorem.Calculating determinants is difficult if n is large. However, we will usuallyonly need the bivariate case∣ a bc d∣ = ad − bcExample 1.6.2.Suppose f is the density on R 2 defined byf(x, y) = 1 ( )2π exp − x22 − y2, (x, y) ∈ R 2 .2Find the joint density of the variablesU = XV = Y/X(This transformation is undefined when X = 0, but that event occurs withprobability zero and may be ignored. We can redefine the sample space toexclude the y-axis without changing any probabilities).The inverse transformation isX = UY = UV