Transformation Examples

Probability · Axiom Academy

EXAMPLE Transformation Examples Finding the PDF of a transformed random variable using the change of variables technique Excellent work! You've transformed a random variable using the change of variables method. Here's what we learned: Inverse Transformation: Express X in terms of Y by solving Y = g(X) for X. This gives us X = g⁻¹(Y). Jacobian Factor: The derivative |dx/dy| accounts for how the transformation stretches or compresses probability density. Transformation Formula: f_Y(y) = f_X(g⁻¹(y)) · |dx/dy| combines the original PDF with the Jacobian to get the new PDF. Domain Transformation: If X ranges from a to b, then Y = 2X + 3 ranges from 2a + 3 to 2b + 3. Always transform the domain! This technique generalizes to any monotonic transformation. For non-monotonic transformations, the method requires additional care with multiple branches.

This is the written version of the interactive lesson above. See the full Probability course.