Chapter 8: Miscellaneous Methods

1. Give the formal derivation of the transformation theorem for continuous random variables.
2. Explain the rationale for the form that the density of \(Y\) takes in terms of the density of \(X\) and the transformation \(Y = g(X)\).
3. State the hypotheses for the transformation theorem for continuous random variables.
4. Use the transformation theorem to determine the probability density function of a random variable \(Y = g(X)\) given the probability density function of \(X\).
5. Give the formal derivation of the transformation theorem for continuous random vectors.
6. Define the Jacobian of a vector-valued function \(\mathbf{g}\) of a vector argument \(\mathbf{x}\).
7. Use the transformation theorem to determine the probability density function of a random vector \(\mathbf{Y} = g(\mathbf{X})\) given the probability density function of \(\mathbf{X}\).
8. Use moment generating functions to determine the distribution of a linear combination of random variables with known joint moment generating function.