stat341 / CM 361: Difference between revisions
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Instructor: Ali Ghodsi | Instructor: Ali Ghodsi | ||
==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 /> | '''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> | 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]\displaystyle{ U~ \sim~ Unif [0,1] }[/math].
Step 2: Compute [math]\displaystyle{ X = F^{-1}(U) }[/math].
Example:
Suppose we want to draw a sample from [math]\displaystyle{ f(x) = \lambda e^{-\lambda x} }[/math] where [math]\displaystyle{ x\gt 0 }[/math].
We need to first find [math]\displaystyle{ F(x) }[/math] and
then [math]\displaystyle{ F^{-1} }[/math].
[math]\displaystyle{ F(x) = \int^x_0 \theta e^{-\theta u} du = 1 - e^{-\theta x} }[/math]
[math]\displaystyle{ F^{-1}(x) = \frac{-log(1-y)}{\theta} }[/math]
Now we can generate our random
sample [math]\displaystyle{ i=1\dots n }[/math] from [math]\displaystyle{ f(x) }[/math] by:
[math]\displaystyle{ 1)\ u_i \sim UNIF(0,1) }[/math]
[math]\displaystyle{ 2)\ x_i = \frac{-log(1-u_i)}{\theta} }[/math]
The [math]\displaystyle{ x_i }[/math] are now a random sample from [math]\displaystyle{ f(x) }[/math].
The major problem with this approach is that we have to find
[math]\displaystyle{ F^{-1} }[/math] and for many distributions it is too difficult to find the inverse of
[math]\displaystyle{ F(x) }[/math].