## Description

Problem 1. (3 points)

(a) (pencil and paper) Suppose that you use the Monte Carlo method to estimate the value

of a given quantity A. One simulation run gives you a single numerical value denoted by

Ai and by running the simulation n times you get the set {Ai}

n

i=1

. How can you obtain a

reliable estimate of the true value of A and how can you calculate an estimate of the error?

How is the central limit theorem related to the estimation?

(b) (computer) Write a program that computes an estimate of π using the “hit-or-miss”

Monte Carlo method. The program should also calculate an error estimate σ (as you

described in part (a)) and the absolute deviation from the correct value (π = 3.141592654).

Do this by generating N uniformly distributed random points inside the square defined by

−1 ≤ x ≤ 1 and −1 ≤ y ≤ 1. Calculate the number of points which are inside the circle

x

2 + y

2 = 1

2

. For each value of N, perform n = 1000 independent measurements. The

resulting average value

πest = π¯ =

1

n

n

∑

i=1

πi

is your MC estimate. The absolute error is ∆ = |πest −π|.

Plot your estimate of π, the error estimate log(σ) and the average absolute error log(∆) as

a function of log(N) for N = 1000−30000 (do a series of runs increasing N by 1000 at

each round). If the calculation takes too long for testing purposes, use smaller values of

N for testing. Plot the error and the error estimate in a single figure. What relation does

the error estimate follow? What about the absolute error?

1

Problem 2. (computer) (0 points – do not hand in)

Write a program which uses the “sample-mean” method to compute the integral

I =

Z 2

0

Z 6

3

Z 1

−1

(yx2 +zlny+e

x

)dx

dy

dz

Include the calculation of an error estimate in your program. Show the results as a function of N (number of random points). The correct answer is I ≈ 49.9213.

Problem 3 (computer) (2 points)

(a) Write a program which uses importance sampling to compute the integral

I =

Z 2

1

(2x

8 −1)dx

by using the weight function w(x) = Cx8

. Plot the result as a function of N.

(b) Compute the same integral using the sample mean method. Plot the absolute error

|Iest −I| as a function of N for the results obtained using the sample mean method and for

the results obtained in (a) using importance sampling. Which method is better?

Problem 4. (computer) (0 points – do not hand in)

Write a program which simulates the process of radioactive decay. You are given a sample

of N = 20000 radioactive nuclei each of which decays at a rate p per second. What is the

half-life of the sample if p = 0.4? Calculate the estimate of the half-life ht1/2

i from

m independent measurements and include an estimation of the error in your program.

Increase m until you reach an accuracy of at least 0.005.

2