Lecture 8

Question 1

The P-value

Answer 1

P-value = 1 - P(χ^2obs) = 1 - χ^2obs ∫ 0 p0 x^(v/2 -1) exp[-x/2] dx = 0.02

Question 2

Monte Carlo Methods

Answer 2

We need methods for generating random variables

Question 3

Uniform random numbers

Answer 3

Generating uniform random numbers, drawn from the pdf U[0,1]

Question 4

Correlations between neighbouring psuedo-random numbers

Answer 4

Sequential patterns in the sampled values would show up as structure in the phase portraits

Question 5

Auto-correlation function

Answer 5

ρ(j) = Σ[(xi-x̄)(xi+j-x̄)]/√Σ(xi-x̄)^2 √Σ(xi+j-x̄)^2

j is known as the lag

Question 6

if the sequence is uniformly random we expect

Answer 6

ρ(j) = 1 for j = 0

ρ(j) = 0 otherwise

Question 7

variable transformations

Answer 7

y = y(x) be monotonic

p(y)dy = p(x)dx

p(y) = p(x(y))/|dy/dx|