Deferred Problems

EE 364: Spring 2026

Daily Derivation: Beta-Binomial Conjugacy

Prove that a beta prior \(h(\theta) = \textrm{Beta}(\alpha, \beta)\) is conjugate to a binomial likelihood \(g(x | \theta) = b(n, \theta)\): \(f(\theta | x) = \textrm{Beta}(\alpha + x, \beta + n - x)\).

Mixture Prior Conjugacy

Suppose Bayesian prior \(p_k(\theta)\) and posterior \(p_k(\theta|x)\) are conjugate with likelihood \(f(x|\theta)\) for \(k = \{1, \ldots, n\}\). Show that the mixture prior \(p(\theta) = \sum_{k=1}^{n} w_k p_k(\theta)\) is conjugate and find the conjugate posterior \(p(\theta|x)\) as a function of \(p_k(\theta|x)\) — mixing weights \(0 < w_k < 1\) and \(\sum_{k=1}^{n} w_k = 1\). Use this fact to find the posterior of the success probability \(p\) after performing twelve coin-flips and observing 9 successes under prior \(p \sim 0.25 \textrm{Beta}(1,1) + 0.75 \textrm{Beta}(5,5)\).

Problem 6

Suppose that there are \(n\) discrete probability density functions \(f_1, f_2, \ldots, f_n\), defined on the same sample space \(\Omega\). A probability density is a nonnegative function on the elements \(k \in \Omega\) that sums to one. Suppose further that the \(n\) nonnegative weights \(w_1, w_2, \ldots, w_n\) also sum to one. Form the weighted mixture \(f\) as follows: \[f(k) = \sum_{j=1}^{n} w_j f_j(k).\] Is the mixture \(f\) a probability density function?

Convergence Under Linear Combination

  1. Random sequences \(X_n \overset{p}{\longrightarrow} X\) and \(Y_n \overset{p}{\longrightarrow} Y\). Show that \(X_n + a Y_n \overset{p}{\longrightarrow} X + a Y\) for any \(a \in \mathbb{R}\). [Hint: triangle inequality.]

  2. Random sequences \(V_n \overset{m}{\longrightarrow} V\) and \(W_n \overset{m}{\longrightarrow} W\). Show that \(V_n + b W_n \overset{m}{\longrightarrow} V + b W\) for any \(b \in \mathbb{R}\). [Hint: Cauchy-Schwarz.]

Expected Distance (Uniform)

A firehouse is to be built at some point along a road with length \(L\). Fires occur uniformly over the road. If the firehouse is built at a distance a from the left endpoint of the road what is the expected distance the firetruck will have to travel?

Problem 18

Computer Problems: Approximate the following integrals using a Monte Carlo simulation. Compare your estimates with the exact values (if known):

  1. \(\displaystyle \int_{-2}^{2} e^{x + x^2} dx\).

  2. \(\displaystyle \int_{0}^{4 \pi} \textrm{sinc}(x) dx\).

  3. \(\displaystyle \int_{0}^{1} \int_{0}^{1} e^{-(x + y)^2} dy dx\).