Another Nice Application of Difference Equations

Here's a nice problem I encountered in a course on applied stochastic processes.

Problem:

Show that the set of all pairs of positive integers can be placed into one-one correspondence with the positive integers by giving an explicit one-one mapping between the two sets.

Solution:

This can be done expediently using the theory of partial difference equations. A standard diagonalization can be characterized by the following relations:
1. $$f(1,1)=1$$
2. $$f(x,y) = f(x-1,y)+x+y$$
3. $$f(x,y) = f(x,y-1)+x+y-1$$
These can be written in difference notation as:
1. $$f(1,1)=1$$
2. $$\frac{\Delta f}{\Delta x} = x+y$$
3. $$\frac{\Delta f}{\Delta y} = x+y-1$$
Equation 2 gives $$\frac{{\Delta}^2 f}{{\Delta x}{\Delta y}} = 1$$ and equation 3 gives $$\frac{{\Delta}^2 f}{{\Delta y}{\Delta x}} = 1,$$ so this is an exact partial difference equation. Summing using equation 2, $f(x,y) = x(x+1)/2 + xy + g(y).$ Differencing and setting this equal to equation 3, $x+\frac{\Delta g}{\Delta y} = x+y-1$ and so $$g(y) = y(y+1)/2 - y + C$$ . Finally $$f(1,1)=1$$ implies that $$C=-1$$ , and so $f(x,y) = x(x+1)/2 + xy + y(y+1)/2 - y - 1.$

A Bayes' Solution to Monty Hall

For any problem involving conditional probabilities one of your greatest allies is Bayes' Theorem. Bayes' Theorem says that for two events A and B, the probability of A given B is related to the probability of B given A in a specific way.

Standard notation:

probability of A given B is written $$\Pr(A \mid B)$$
probability of B is written $$\Pr(B)$$

Bayes' Theorem:

Using the notation above, Bayes' Theorem can be written: $\Pr(A \mid B) = \frac{\Pr(B \mid A)\times \Pr(A)}{\Pr(B)}$Let's apply Bayes' Theorem to the Monty Hall problem. If you recall, we're told that behind three doors there are two goats and one car, all randomly placed. We initially choose a door, and then Monty, who knows what's behind the doors, always shows us a goat behind one of the remaining doors. He can always do this as there are two goats; if we chose the car initially, Monty picks one of the two doors with a goat behind it at random.

Assume we pick Door 1 and then Monty sho…

Notes on Setting up a Titan V under Ubuntu 17.04

I recently purchased a Titan V GPU to use for machine and deep learning, and in the process of installing the latest Nvidia driver's hosed my Ubuntu 16.04 install. I was overdue for a fresh install of Linux, anyway, so I decided to upgrade some of my drives at the same time. Here are some of my notes for the process I went through to get the Titan V working perfectly with TensorFlow 1.5 under Ubuntu 17.04.

Old install:
Ubuntu 16.04
EVGA GeForce GTX Titan SuperClocked 6GB
2TB Seagate NAS HDD

New install:
Ubuntu 17.04
Titan V 12GB
/ partition on a 250GB Samsung 840 Pro SSD (had an extra around)
/home partition on a new 1TB Crucial MX500 SSD
New WD Blue 4TB HDD