# Random triangles

In this exercise, we will estimate, by running simulations, the
expected area of a certain randomly selected triangle. Given a fixed
deterministic triangle, we will uniformly and independently at random
select three points inside it, and consider the area of the triangle
formed by these three random points. It is known that the expected area
of such a random triangle is \(\tfrac{1}{12}\) that of the original
triangle.

## Warm up

Consider the triangle with vertices: \((0,0)\), \((3,0)\), and \((1,2)\). One way to select a point
uniformly at random in this triangle, is to uniformly select one at
random on a square containing this triangle: if the point lands on the
triangle, then we take it, otherwise, we try again. Code this
procedure.

## The problem

Consider a fixed triangle of your choosing; you can use the previous
one. Simulate the independent uniform points on the triangle, and
compute its area. You may find this
helpful, if you are at a loss of how to compute the area. Finally repeat
this procedure, and find the average value. Enjoy!

# Linear regression

In this more open ended exercise, we consider the problem of simple
linear regression, and how you might use R (or Python) to illustrate to
a someone, taking a first module in statistic, that the usual least
squared estimators are good.

## Some basic defintions that might be useful

Let \(x_1, \ldots, x_n, \ y_1, \ldots, y_n
\in \mathbb{R}\). Define \[\overline{xy} = \frac{1}{n} \sum_{i=1}^n x_i
y_i,\] \[ \overline{x^2}
= \frac{1}{n}\sum_{i=1}x_i^2,\] \[V(x) = \frac{1}{n} \sum_{i=1}^n(x_i -
\bar{x})^2.\] \[CV(x,y) = \frac{1}{n}
\sum_{i=1}^n(x_i - \bar{x})(y_i - \bar{y}). \] Consider the
linear model, \(y_i = mx_i + b +
\epsilon_i\), where \(\epsilon_i\) are iid mean zero random
variables, and \((m,b)\) are unknown.
In your previous courses, with some calculus, you derived an estimator
for the slope:

\[\hat{m}
= CV(x,y) / V(x).\]

Demonstrate with by simulations that this is a good estimator.

### Endnotes

Version: 08 October 2023