Difference between revisions of "stat341 / CM 361"

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Instructor: Ali Ghodsi  
 
Instructor: Ali Ghodsi  
  
<math>Insert formula here</math>
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==Sampling (Generating Random numbers)==
 
==Sampling (Generating Random numbers)==
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Step 1: Draw <math> U~ \sim~ Unif [0,1] </math>. <br />
 
Step 1: Draw <math> U~ \sim~ Unif [0,1] </math>. <br />
 
Step 2: Compute <math> X = F^{-1}(U) </math>.<br />
 
Step 2: Compute <math> X = F^{-1}(U) </math>.<br />
'''Example:'''<br />
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'''Example:'''<br />Suppose  we want to draw a sample from <math> f(x) =  \lambda e^{-\lambda x} </math> where <math>x>0</math>. <br />We need to first find <math>F(x)</math> and
Suppose  we want to draw a sample from <math> f(x) =  \lambda e^{-\lambda x} <\math>
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then <math>F^{-1}</math>.<br />
 +
<math> F(x) = \int^x_0 \theta e^{-\theta u} du = 1 - e^{-\theta x} </math> <br />
 +
<math> F^{-1}(x) = \frac{-log(1-y)}{\theta} </math> <br />
 +
Now we can generate our random
 +
sample <math>i=1\dots n</math> from <math>f(x)</math> by:<br />
 +
<math>1)\ u_i \sim UNIF(0,1) </math><br />
 +
<math>2)\ x_i = \frac{-log(1-u_i)}{\theta} </math><br />
 +
The <math>x_i</math> are now a random sample from <math>f(x)</math>. <br />
 +
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>.

Revision as of 08:45, 13 May 2009

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].
Example:
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].