- Apply the Monte Carlo method using 2000 independent realisations of a
U[0,1] variable to estimate the value of the definite integral
2 1.96
2
1.96
1 sin( ) 2
x
x e dx
π
§ ·
¨ ¸ ¨ ¸ © ¹
ª º
« »
¬ ¼ ³ .
Clearly state any changes of variable required and derive the resulting
equivalent definite integral. - Consider the problem of evaluating the value of the definite integral
2 2
0
cos( ) x e x dx
f
³ .
i) Using the substitution 1 x y e and 5000 independent
realisations of a U[0,1] variable, estimate the value of the integral.
ii) Find an alternative suitable substitution and obtain a second
estimate of the value of the integral using the same 5000
realisations of a U[0,1] variable.
Clearly state your change of variable and derive the resulting
equivalent definite integral.
3.
Consider the problem of evaluating the value of the definite integral
0
2
10
1
dx
x f
« »
« » ³ ¬ ¼ .
i) Using the substitution 2
1
1 y
x
and 5000 independent
realisations of a U[0,1] variable, estimate the value of the integral.
Hint: Think carefully when calculating x y( )for the required substitution.
ii) Find an alternative suitable substitution and obtain a second
estimate of the value of the integral using the same 5000
realisations of a U[0,1] variable.
Clearly state your change of variable and derive the resulting
equivalent definite integral.
Note: z denotes the floor function which, for any z , takes the
value of the largest integer z . For example, if z 3.5, then
2
10 10 0
1 ( 3.5) 13.25
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