11 - TRANSFORMING DENSITY FUNCTIONS It can be expedient to use a transformation function to transform one probability density function into another. As an introduction to this topic, it is helpful to recapitulate the method of integration by substitution of a new variable. Integration by Substitution of a new Variable Imagine that a newcomer to integration comes across the following: 2 2x cos x2 dx 0 Assuming that the newcomer doesn't notice that the integrand is the derivative of sin x2, one way to proceed would be to substitute a new variable y for x2: Let y = x2 Replace the limits x = 0 and x = / by y = 0 and y = { Replace 2x cos x2 by 2 /y cos y Note that x = Vy and hence dy = = and so replace dx by day- 2 y The original problem is thereby transformed into the following integration: 0 cos y dy = siny = 1 The General Case It is instructive to develop the general case alongside the above example: General Case ·b f (x) dx a Choose a transformation function y (x) Note its inverse x(y) - 11.1 - Replace the limits by y(a) and y(b) 0 and 1 Replace f (x) by f (x(y)) 2 /y cos y Replace dx by au dy ry(b) dx - dy Result is Jy(a) f(x(y)) dy Above Example 2 2x cos x2 dx 0 y(x) = x2 x(y) = V? 1 dy 2 /y F 2 cos y dy 0
Application to Probability Density Functions The previous section informally leads to the general formula for integration by substitution of a new variable: a f(x) dx = [0 (x ) dx dy dy (11.1) This formula has direct application to the process of transforming probability density functions ... Suppose X is a random variable whose probability density function is f (x). By definition: P(a 6 X < b) = [" a f (x) dx (11.2) Any function of a random variable is itself a random variable and, if y is taken as some transformation function, y(X) will be a derived random variable. Let Y = y(X). Notice that if X = a the derived random variable Y = y(a) and if X = b, Y = y(b). Moreover, (subject to certain assumptions about y) if a 6 X < b then y(a) 6 Y < y(b) and P(y(a) 6 Y < y(b))=P(a 6X<b). Hence, by (11.2) and (11.1): P(y(a) 6 Y < y(b)) = P(a 6 X < b) = / b dy (11.3) a f (x) dx = cy(b) Jy(a) = from () dy dx Notice that the right-hand integrand f (x(y)) dx dy is expressed wholly in terms of y. Calling this integrand g(y): P(y(a) 6Y < y(b)) = Jy(a) ry(b) g(y) dy This demonstrates that g(y) is the probability density function associated with Y . The transformation is illustrated by the following figures in which the function f (x) (on the left) is