1. Basic Optimization, MLE for IID Poisson Data

Suppose \(y_i\) is a count. That is \(y_i \in {0,1,2,....}\).

In this case, a very common model is to assume the Poisson disttribuion: \[ P(Y=y \;|\; \lambda) = \frac{e^{-\lambda} \, \lambda^y}{y!}, \; y = 0,1,2,\ldots \]

Given \(Y_i \sim Poisson(\lambda)\) iid, (that is, \(Y_i = y_i\)), what is the MLE of \(\lambda\)?

2. Constrained Optimization

Let \[ f(x) = (x_1 - a_1)^2 + (x_2 - a_2)^2, \;\; g(x_1,x_2) = x_1^2 + x_2^2 - 1. \]

Minimize \(f(x)\) subject to the constraint that \(g(x) \leq 0\).

First draw simple pictures to make the solution obvious.

Then check that the lagrange multiplier first order condition conforms with with your intution.

How does the norm of \((a_1,a_2)\) affect the solution !!??