# Difference between revisions of "stat341 / CM 361"

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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 07:45, 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].