Example 5.  Use Monte Carlo simulation to approximate the integral  [Graphics:Images/MonteCarloPiMod_gr_137.gif].  

Solution 5.

Set up the function and the rectangular box enclosing the area.

[Graphics:../Images/MonteCarloPiMod_gr_138.gif]


[Graphics:../Images/MonteCarloPiMod_gr_139.gif]

[Graphics:../Images/MonteCarloPiMod_gr_140.gif]

Explore what happens with  [Graphics:../Images/MonteCarloPiMod_gr_141.gif]  points.

 

 

 

[Graphics:../Images/MonteCarloPiMod_gr_142.gif]

[Graphics:../Images/MonteCarloPiMod_gr_143.gif]

[Graphics:../Images/MonteCarloPiMod_gr_144.gif]

 

 

 

[Graphics:../Images/MonteCarloPiMod_gr_145.gif]

[Graphics:../Images/MonteCarloPiMod_gr_146.gif]

[Graphics:../Images/MonteCarloPiMod_gr_147.gif]

 

 

 

[Graphics:../Images/MonteCarloPiMod_gr_148.gif]

[Graphics:../Images/MonteCarloPiMod_gr_149.gif]

[Graphics:../Images/MonteCarloPiMod_gr_150.gif]

 

 

We are done.  

Aside.  The analytic value of the integral can be found.

[Graphics:../Images/MonteCarloPiMod_gr_151.gif]



[Graphics:../Images/MonteCarloPiMod_gr_152.gif]

[Graphics:../Images/MonteCarloPiMod_gr_153.gif]

 

 

 

We are done.  

Aside.  This is the standard normal distribution function.  Suppose that the interval of integration is [Graphics:../Images/MonteCarloPiMod_gr_154.gif].

[Graphics:../Images/MonteCarloPiMod_gr_155.gif]


[Graphics:../Images/MonteCarloPiMod_gr_156.gif]

[Graphics:../Images/MonteCarloPiMod_gr_157.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005