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Friday, November 16, 2012

Fun with Uniform Random Numbers

Q: You have two uniformly random numbers x and y (meaning they can take any value between 0 and 1 with equal probability). What distribution does the sum of these two random numbers follow? What is the probability that their product is less than 0.5.

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists


A: Let z = x + y be the random variable whose distribution we want. Clearly z runs from 0 to 2. Let 'f' denote the uniform random distribution between [0,1]. An important point to understand is that f has a fixed value of 1 when x runs from 0 to 1 and its 0 otherwise.
So the probability density for z, call it P(z) at any point is the product of f(y) and f(z-y), where y runs from 0 to 1.


However in that range f(y) is equal to 1. So the above equation becomes

From here on, it gets a bit tricky. Notice that the integral is a function of z. Let us take a look at how else we can simply the above integral.

It is easy to see that f(z-y) = 1 when (z-y) is between [0,1]. This is the same as saying
Likewise, f(z-y) = 1 when y is lesser than z and greater than 0. ie

Combining the two cases above results in a discontinuous function as

which is a triangular function.

 Now that we done with the sum, what about the product xy? A quick way to go about it is to visualize a 2 dimensional plane. All the points (x,y) within the square [0,1]x[0,1] fall in the candidate space. The case when xy = 0.5 makes a curve
 The area under the curve would represent the cases for which xy <= 0.5 (shown shaded below). Since the area for the square is 1, that area is the sought probability.

The curve intersects the square at [0.5,1] and [1,0.5]. The area under the curve would the be sum of the 2 quadrants (1/4 each) along with the integral of y = 0.5/x under the range 0.5 to 1 yielding
$$
\begin{align*}
P(xy\lt 0.5) &= \frac{1}{2} + \int_{0.5}^{1}\frac{0.5}{x}dx \\
&= \frac{1}{2} + \frac{1}{2}ln2 \approx 0.85
\end{align*}
$$

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Here are a few
Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.

Introduction to Algorithms
This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists

Introduction to Probability Theory

An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)

Introduction to Probability, 2nd Edition

The Mathematics of Poker
Good read. Overall Poker/Blackjack type card games are a good way to get introduced to probability theory

Let There Be Range!: Crushing SSNL/MSNL No-Limit Hold'em Games
Easily the most expensive book out there. So if the item above piques your interest and you want to go pro, go for it.

Quantum Poker
Well written and easy to read mathematics. For the Poker beginner.


Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
An excellent resource (students/engineers/entrepreneurs) if you are looking for some code that you can take and implement directly on the job.

Understanding Probability: Chance Rules in Everyday Life A bit pricy when compared to the first one, but I like the look and feel of the text used. It is simple to read and understand which is vital especially if you are trying to get into the subject

Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems) This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.


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