Transformations of Random Variables for Exam P

CDF Method

The CDF method works for any transformation Y = g(X). First find F_Y(y) = P(Y <= y) = P(g(X) <= y), then differentiate to get f_Y(y). This is the most general technique and should be your default approach on Exam P.

Example: If X ~ Uniform(0, 1) and Y = -ln(X), then F_Y(y) = P(-ln(X) <= y) = P(X >= e^(-y)) = 1 - e^(-y) for y > 0. Differentiating gives f_Y(y) = e^(-y), so Y ~ Exponential(1). This result is the basis of inverse transform simulation.

Change of Variable Formula

For a one-to-one differentiable transformation Y = g(X) with inverse X = g^(-1)(Y) = h(Y), the PDF of Y is f_Y(y) = f_X(h(y)) * |h'(y)|. The absolute value of the Jacobian ensures the density is non-negative regardless of whether the transformation is increasing or decreasing.

For two-dimensional transformations (U, V) = g(X, Y), the joint density of (U, V) involves the absolute value of the Jacobian determinant: |det(partial derivatives of the inverse transformation)|.

Common Exam P Transformations

Know these results: if X ~ Exponential(lambda), then cX ~ Exponential(lambda/c) for c > 0. If X ~ Normal(mu, sigma^2), then aX + b ~ Normal(a*mu + b, a^2*sigma^2). If X ~ Gamma(alpha, lambda), then cX ~ Gamma(alpha, lambda/c). If U ~ Uniform(0, 1), then F^(-1)(U) has CDF F (the probability integral transform). These shortcuts save time on exam day.