# So one can estimate π by g x algorithm i generate u

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. So one can estimate π by ¯ g ( X ) . Algorithm: I Generate u 1 , · · · , u n from U (0 , 1) I Calculate g ( u 1 ) , · · · , g ( u n ) . I Estimate π by ¯ g = n i =1 g ( u i ) /n . I Construct a 95% CI for π : denote s g = 1 n - 1 n i =1 [ g ( u i ) - ¯ g ] 2 , ¯ g ± 1 . 96 s g / n In general, R b a g ( x ) dx = ( b - a ) R b a g ( x ) b - a dx = ( b - a ) E ( g ( X )) , where X U [ a, b ] , hence can be estimated by ( b - a ) g ( X ) Jimin Ding, Math WUSTL Math 494 Spring 2018 5 / 8

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Example 3: Generate a r.v. from F - 1 ( U ) Theorem 4.8.1 +Ex 4.8.1: If X F , then F ( X ) U [0 , 1] . And Y = F - 1 ( U ) F . Jimin Ding, Math WUSTL Math 494 Spring 2018 6 / 8
Example 3: Generate a r.v. from F - 1 ( U ) Theorem 4.8.1 +Ex 4.8.1: If X F , then F ( X ) U [0 , 1] . And Y = F - 1 ( U ) F . For example, to generate X from f X ( x ) = exp { x - e x } , x R : (Extreme value distribution, log of Weibull distribution) Jimin Ding, Math WUSTL Math 494 Spring 2018 6 / 8

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Example 3: Generate a r.v. from F - 1 ( U ) Theorem 4.8.1 +Ex 4.8.1: If X F , then F ( X ) U [0 , 1] . And Y = F - 1 ( U ) F . For example, to generate X from f X ( x ) = exp { x - e x } , x R : (Extreme value distribution, log of Weibull distribution) It is easy to show the cdf is F X ( x ) = 1 - exp {- e x } , x R Jimin Ding, Math WUSTL Math 494 Spring 2018 6 / 8
Example 3: Generate a r.v. from F - 1 ( U ) Theorem 4.8.1 +Ex 4.8.1: If X F , then F ( X ) U [0 , 1] . And Y = F - 1 ( U ) F . For example, to generate X from f X ( x ) = exp { x - e x } , x R : (Extreme value distribution, log of Weibull distribution) It is easy to show the cdf is F X ( x ) = 1 - exp {- e
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• Fall '08
• Staff
• Monte Carlo method, Monte Carlo methods in finance, Markov chain Monte Carlo, Jimin Ding

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