Consider the function \(x \to \frac{1}{\sqrt{2\pi}} e^{-x^2/2}\). Suppose that I want to sample a point uniformly under the area enclosed by the function and the line \(y=0\). How would you accomplish this? Code it.
Let \(\epsilon_1, \epsilon_2\) and \(\epsilon_3\) be independent random variables. Let \(\phi_1, \phi_2\), \(\phi_3\) be some deterministic functions. Set \(X_1 = \phi_1(\epsilon_1)\), \(X_2 = \phi_2(X_1, \epsilon_2)\), and \(X_3 = \phi_3(X_2,\epsilon_3)\). Assume that \(X_1, X_2\), and \(X_3\) are discrete random variables. It maybe helpful to visualize this arrangement as \[X_1 \to X_2 \to X_3.\]
Suppose that \(\mathbb{P}(X_3=x_3, X_2=x_2, X_1=x_1) >0\).
\[\mathbb{P}(X_3 = x_3 | X_2=x_2, X_1=x_1) = \mathbb{P}(X_3 = x_3 | X_2=x_2).\]
\[\mathbb{P}(X_3=x_3, X_1=x_1 | X_2=x_2) = \mathbb{P}(X_3=x_3| X_2=x_2)\mathbb{P}(X_1=x_1 | X_2=x_2)\]
Let \(\epsilon_1, \epsilon_2\) and \(\epsilon_3\) be independent random variables. Let \(\phi_1, \phi_2\), \(\phi_3\) be some deterministic functions. Set \(X_1 = \phi_1(\epsilon_1)\), \(X_2 = \phi_2(X_1,\epsilon_2)\), and \(X_3 = \phi_3(X_1, \epsilon_3)\). Assume that \(X_1, X_2\), and \(X_3\) are discrete random variables.
Are \(X_2\) and \(X_3\) conditionally independent given \(X_1\)? Explain.
Let \(\epsilon_1, \epsilon_2\) and \(\epsilon_3\) be independent random variables. Let \(\phi_1, \phi_2\), \(\phi_3\) be some deterministic functions. Set \(X_1 = \phi_1(\epsilon_1)\), \(X_2 = \phi_2(\epsilon_2)\), and \(X_3 = \phi_3(X_1, X_2,\epsilon_3)\). Assume that \(X_1, X_2\), and \(X_3\) are discrete random variables. Are \(X_1\) and \(X_2\) conditionally independent given \(X_3\)? Explain.
Let \(X = (X_i)_{i=0}^{\infty}\) be a Markov chain on the state space \(S\). Let \(\phi: S \to S\) be deterministic function. Suppose that \(Y_i = \phi(X_i)\). Is \(Y=(Y_0,Y_1,\ldots)\) always a Markov chain? Explain.