Stat 5101 Lecture Notes - School of Statistics

1.2. CHANGE OF VARIABLES 11Every invertible functionSg−→ Thas an inverse functionT −→ g−1S(note g −1 goes in the direction opposite to g) satisfyingg ( g −1 (y) ) = y,y = inTandg −1( g(x) ) = x, x = inS.A way to say this that is a bit more helpful in doing actual calculations isy = g(x)wheneverx = g −1 (y).The inverse function is discovered by trying to solvefor x. For example, ify = g(x)y = g(x) =x 2thenx = √ y = g −1 (y).If for any y there is no solution or multiple solutions, the inverse does not exist (ifno solutions the function is not onto, if multiple solutions it is not one-to-one).Change of Variable for Invertible TransformationsFor invertible transformations Theorem 1.2 simplifies considerably. The setB in the theorem is always a singleton: there is a unique x such that y = g(x),namely g −1 (y). SoB = { g −1 (y) },and the theorem can be stated as follows.Theorem 1.3. If X is a discrete random variable with density f X and samplespace S, ifg:S→T is an invertible transformation, and Y = g(X), then Y isa discrete random variable with density f Y defined byf Y (y) =f X(g −1 (y) ) , y ∈ T. (1.15)Example 1.2.3 (The “Other” Geometric Distribution).Suppose X ∼ Geo(p), meaning that X has the densityf X (x) =(1−p)p x , x =0,1,2,... (1.16)Some people like to start counting at one rather than zero (Lindgren amongthem) and prefer to call the distribution of the random variable Y = X + 1 the