MTH3007B Weekly Problems 7

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7.1. MC Estimate of a One-Dimensional Integral

Question

Approximate the following expression involving an integral using your own routines for Monte Carlo integration:

$5{\int }_0^{\pi /2}\cos (x)\mathrm{d}x.$

Also determine an error estimate for this quantity.

For this part, use $100000$ random numbers in your Monte Carlo routine. What is your numerical estimate for the expression?

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7.2. Standard Deviation Error Estimate

Question

Using the same Monte Carlo computation as in 7.1. MC Estimate of a One-Dimensional Integral, what is your estimate of the error (one standard deviation) in the expression

$5{\int }_0^{\pi /2}\cos (x)\mathrm{d}x$

when you use $100000$ random numbers in your Monte Carlo integration?

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7.3. MC Approximation of a Four-Dimensional Integral

Question

Approximate the following four-dimensional integral using your own routines for Monte Carlo integration:

${\int }_0^3{\int }_0^1{\int }_0^1{\int }_0^1\frac{\mathrm{d}A\mathrm{d}B\mathrm{d}C\mathrm{d}D}{1+A+B+2C^2+D^3}.$

In doing so, sample the function

$\frac{1}{1+A+B+2C^2+D^3}$

exactly $100000$ times using Monte Carlo integration. What is your numerical estimate for the value of the integral?

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