Week 6 Monte Carlo Basics

Question 1

how are measurements typically represented

Answer 1

using its central value and its accompanying uncertainty

Question 2

define the two types of error

Answer 2

statistical error - errors that arise from random fluctuation in the experiment
systematic error - errors that are consistent and constant for all data in an experiment and therefore the effect can’t be reduced by repeating the experiment

Question 3

what are Monte Carlo methods

Answer 3

class of computational techniques that rely on using random sampling in order to solve problems

Question 4

what are the 3 scenarios you would use Monte carlo

Answer 4

numerical integration
sampling from an arbitrary probability distribution
optimisation

Question 5

what is the starting point for most monte carlo methods

Answer 5

use a sequence of uniformly distributed random numbers

Question 6

what is the name of a commonly used pseudo-random number generator

Answer 6

linear congruential generator

LCG

Question 7

what is the LCG equation

Answer 7

ni = (an_i-1 + c)mod(m)

Question 8

what is the transformation used to obtain a uniform distribution

Answer 8

ri = ni / m

Question 9

what is the multiplicative congruential generator

Answer 9

a special LCG case where the incremental value c is c=0

Question 10

what is the procedure of crude monte carlo integration

Answer 10

generate n random numbers from a uniform distribution

evaluate the integral according to b-a/n Σf(xi)

Question 11

what is the procedure for rejection sampling integration

Answer 11

generate a random number xi that is uniformly distributed between two limits a and b
generate a second random variable number ui
if ui satisfies ui < f(xi) / fmax then accept xi

Question 12

what are the three advantages of Monte carlo integration

Answer 12

simplicity
unbiased
parallel by nature

Question 13

what are the two disadvantages of monte carlo integration

Answer 13

slow rate of convergence

difficult to evaluate the uncertainty of the approximation

Question 14

what are the 3 methods to generate a sample from an arbitrary probability distribution

Answer 14

Weighted sampling
Rejection sampling
Inverse transform sampling

Question 15

how does weighted sampling work

Answer 15

generate a uniform random variable and weight each event by the factor w = f(x)

Question 16

how does inverse transform sampling work

Answer 16

sample a value from a uniform function and substitute into the inverse transform sampling equation to generate samples that follow the distribution f(x)

Question 17

what is the inverse transform sampling equation

Answer 17

x(r) = -rlog(1-r)

Question 18

what is the goal of optimisation

Answer 18

to simulate multiple alternate realisation of a situation and determine which one of them provides the most optimal solution