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Computational Statistics and Data Analysis is a course offered at the University of Waterloo
Spring 2009
Instructor: Ali Ghodsi

Sampling (Generating Random numbers)

Lecture of May 12th, 2009

In order to study statistics computationally, we need a good way to generate random numbers from various distributions using computational methods, or at least numbers whose distribution appears to be random (pseudo-random). Outside a computational setting, this is fairly easy (at least for the uniform distribution). Rolling a die, for example, produces numbers with a uniform distribution very well.

We begin by considering the simplest case: the uniform distribution.

One way to generate pseudorandom numbers is using the Multiplicative Congruential Method. This involves three integer parameters a, b, and n, and a seed variable x0. This method deterministically (based on the seed) generates a sequence of numbers with a seemingly random distribution (with some caveats). It proceeds as follows:

[math]x_{i+1} = ax_{i} + b \mod{n}[/math]

For example, with a = 13, b = 0, m = 31, x0 = 1, we have:

[math]x_{i+1} = 13x_{i} \mod{31}[/math]


[math]\begin{align} x_{0} &{}= 1 \end{align}[/math]
[math]\begin{align} x_{1} &{}= 13*1 + 0 \mod{31} \\ &{}= 13 \end{align}[/math]
[math]\begin{align} x_{2} &{}= 13*13 + 0 \mod{31} \\ &{}= 14 \end{align}[/math]


Inverse Transform Method

Step 1: Draw [math] U~ \sim~ Unif [0,1] [/math].
Step 2: Compute [math] X = F^{-1}(U) [/math].
Suppose we want to draw a sample from [math] f(x) = \lambda e^{-\lambda x} [/math] where [math]x\gt 0[/math].
We need to first find [math]F(x)[/math] and then [math]F^{-1}[/math].

[math] F(x) = \int^x_0 \theta e^{-\theta u} du = 1 - e^{-\theta x} [/math] 

[math] F^{-1}(x) = \frac{-log(1-y)}{\theta} [/math]
Now we can generate our random sample [math]i=1\dots n[/math] from [math]f(x)[/math] by:

[math]1)\ u_i \sim UNIF(0,1) [/math]
[math]2)\ x_i = \frac{-log(1-u_i)}{\theta} [/math]

The [math]x_i[/math] are now a random sample from [math]f(x)[/math].
The major problem with this approach is that we have to find [math]F^{-1}[/math] and for many distributions it is too difficult to find the inverse of [math]F(x)[/math].