Consider the problem of generating samples from \(Y \sim \operatorname{Gamma}(2,10)\).

(a) Direct simulation: Let \(U_{1}, U_{2} \sim\) idd \(\mathscr{U}(0,1)\). Show that \(-\ln \left(U_{1}\right) / 10-\ln \left(U_{2}\right) / 10 \sim\) Gamma \((2,10)\). [Hint: derive the distribution of \(-\ln \left(U_{1}\right) / 10\) and use Example C.1.]

(b) Simulation via MCMC: Implement an independence sampler to simulate from the Gamma \((2,10)\) target pdf \[ f(x)=100 x \mathrm{e}^{-10 x}, x \geqslant 0 \]
using proposal transition density \(q(y \mid x)=g(y)\), where \(g(y)\) is the pdf of an \(\operatorname{Exp}(5)\) random variable. Generate \(N=500\) samples, and compare the true cdf with the empirical cdf of the data.