Uniform Distribution

$X\sim Uniform(a,b)$ if

$P(X=x)=f(x)=\{\begin{array}{ll}\frac{1}{b-a}& a\le x\le b\\ 0& \text{otherwise}\end{array}$

• $E(X)=\frac{b+a}{2}$
• $Var(X)=\frac{1}{12}(b-a{)}^2$.

This distribution may seem simple, but it has two important properties:

1. Gateway to more complex Distribution
   1. You’ve seen this first-hand at Ericsson (more details below)
2. Universality Property
   • Suppose $X$ is a r.v. with CDF $F(X)$. If we are interested in generating $n$ simulations of $X$, i.e. we want to generate $n$ outcomes $\{x_1,x_2,\dots ,x_n\}$, we can generate the following: $\{F^{-1}(u_1),F^{-1}(u_2),\dots ,F^{-1}(u_n)\}$, where $u_1,u_2,\dots ,u_n\sim Uniform(0,1)$

Ericsson: Generating Random Uniform Distributions

This was a tough problem that I faced when working at Ericsson. The built-in rand() function in C seemed to be too slow. However, the faster version using seeds really wasn’t good enough when I tried to use it in Thompson Sampling.

Actually, it seems that they are trying to implement a version of the xorshift. https://en.wikipedia.org/wiki/Xorshift

Let’s look into some alternatives:

• https://stackoverflow.com/questions/1640258/need-a-fast-random-generator-for-c
• http://www.cse.yorku.ca/~oz/marsaglia-rng.html
• From the boost library: https://www.boost.org/doc/libs/1_79_0/doc/html/boost_random/performance.html