stat341 / CM 361

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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)

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].