Composition of Transformations The symbol for a composition of transformations (or functions) is an open circle A notation such as is read as "a translation of (x, y) → (x 1, y 5) after a reflection in the line y = x" You may also see the notation written as This process must be done from right to leftWe create a new random variable Y as a transformation of X For example, Y = X2 We wish to nd the distribution of Y or fY (y) Two techniques we will discuss for continuous rv's (1) Distribution function (cdf) technique (2) Change of variable (Jacobian) technique 1574 Evaluate a triple integral using a change of variables Recall from Substitution Rule the method of integration by substitution When evaluating an integral such as ∫3 2x(x2 − 4)5dx, we substitute u = g(x) = x2 − 4 Then du = 2xdx or xdx = 1 2du and the limits change to u = g(2) = 22 − 4 = 0 and u = g(3) = 9 − 4 = 5 Exponential Function Formula Y=e^x transformations
