Chapter 8: Miscellaneous Methods

1. Use the CDF method to find the distribution of a nonlinear transformation \(Z = g(X, Y)\) of a random vector \((X, Y)\) with known probability density function.
2. Construct a continuous random variate \(X\) with a non-uniform distribution \(F_{X}\) from a continuous random variate \(U\) with a uniform distribution on \((0, 1)\).
3. Give a derivation of the inverse function theorem using the chain rule.
4. Find the range of a random vector \((Z, W) = \mathbf{g}(X, Y)\) that is a transformation of a random vector \((X, Y)\) with known range by “mapping boundaries”.
5. Find the distribution of a linear combination \(Z = a X + b Y\) of the components of a random vector \((X, Y)\) via the introduction of an auxiliary random variable \(W = X\) or \(W = Y\) and then marginalization.