Transformations of Random Variables

Probability · Axiom Academy

LESSON Transformations of Random Variables Finding the distribution of functions of random variables Given: A random variable X with known PDF f X (x). Question: If Y = g(X) for some function g, what is the PDF of Y? We cannot simply substitute Y into f X . The transformation changes the shape and scale of the distribution. 2. Change of Variables Formula For a monotonic (strictly increasing or decreasing) transformation Y = g(X), the PDF of Y is: Here, g -1 is the inverse function and |dg -1 /dy| is the absolute value of its derivative. The derivative term accounts for how the transformation stretches or compresses probability density. 3. Multivariate Transformations and Jacobian For transformations of multiple variables (Y 1 , Y 2 ) = g(X 1 , X 2 ), we use the Jacobian matrix. The Jacobian determinant measures how the transformation scales area elements in the plane. The formula generalizes the single-variable case to higher dimensions.

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