Question

Let 𝐼 = 0∫10 𝑓(𝑥)𝑑x . You are asked to approximate the value of I using (i) uniform random 0 10 ∫ 𝑓(𝑥)𝑑𝑥 variable, (ii) Monte Carlo simulation and (iii) antithetic variates

a. Write down the procedures for the Monte Carlo simulation without using antithetic variates.

b. Write down the antithetic variable.

Answer 1

ii)

Monte Carlo approximation of the integral:

$\int f(x)dx=\frac{1}{n}\sum f(x_i)$

where x_i are are independent observations of X

if n = 10, then:

$\int^{10}_0 f(x)dx=\frac{1}{10}\sum f(x_i)$

iii)

 antithetic variates:

$\int f(x)dx=\frac{1}{n}\sum \frac{f(x_i)}{2}+\frac{1}{n}\sum \frac{f(1-x_i)}{2}$

if n = 10, then:

$\int^{10}_0 f(x)dx=\frac{1}{10}\sum \frac{f(x_i)}{2}+\frac{1}{10}\sum \frac{f(1-x_i)}{2}$

i)

using uniform random variable:

$\int^{b}_a f(x)dx=\frac{b-a}{n}\sum f(x_i)$

$\int^{10}_0 f(x)dx=\frac{10}{n}\sum f(x_i)$

a.

Monte Carlo simulation without using antithetic variates:

$\int f(x)dx=\frac{1}{n}\sum f(x_i)$

where x_i are are independent observations of X

if n = 10, then:

$\int^{10}_0 f(x)dx=\frac{1}{10}\sum f(x_i)$

b.

antithetic variable is $(1-x_i)$