Definitions and Explanations

Question 1

poisson statistics

Answer 1

the number of photons arriving at our detector from a given source will fluctuate

treat the arrival rate of photons statistically

Question 2

poisson statistics assumptions

Answer 2

1. Photons arrive independently in time
2. Average photon arrival rate is constant

Question 3

as Rτ increases

Answer 3

the shape of the Poisson distribution becomes more symmetrical

tends to a normal or gaussian

Question 4

variance

Answer 4

is a measure of the spread in the Poisson distribution

Question 5

Laplace’s basis for plausible reasoning

Answer 5

Probability measures our degree of belief that something is true

Question 6

probability density function

Answer 6

when measuring continuous variables which can take on infinitely many possible values

Question 7

sketch of a poisson PDF

Answer 7

see notes

Question 8

sketch of a uniform PDF

Answer 8

see notes

Question 9

sketch of a central/normal or gaussian pdf

Answer 9

see notes

Question 10

sketch of a cumulative distribution function (CDF)

Answer 10

see notes

Question 11

the 1st moment is called

Answer 11

the mean or expectation value

Question 12

the 2nd moment is called

Answer 12

the mean square

Question 13

the median divides

Answer 13

the CDF into two equal halves

Question 14

the mode is

Answer 14

the value of x for which the pdf is a maximum

Question 15

central limit theorem

Answer 15

explains the importance of a normal pdf in statistics

but still based on the asymptotic behaviour of an infinite ensemble of samples that we didn’t actually observe

Question 16

Isoprobability contours for the bivariate normal pdf

Answer 16

p > 0 : positive correlation y tends to increase as x increases

p < 0 : negative correlation y tends to decrease as x increases

Question 17

as |p| -> 1

Answer 17

contours become narrower and steeper

Question 18

the principle of maximum likelihood

Answer 18

is a method to estimate the parameters of a distribution which fit the observed data

Question 19

if we obtain a very small P-value

Answer 19

we can interpret this as providing little support for the null hypothesis,

which we may then choose to reject

Question 20

Monte-carlo methods

Answer 20

method for generating random variables

Question 21

can test psuedo-random numbers for randomness in several ways

Answer 21

a) histogram of sampled values
b) correlations between neighbouring pseudo-random numbers
c) autocorrelation
d) chi squared

Question 22

Markov Chain Monte Carlo

Answer 22

method for sampling from PDFs

1. start off at some randomly chosen value (a(1),b(1))
2. compute L(a(1),b(1)) and gradient
3. Move in direction of steepest +ve gradient
4. repeat from step 2 until (a(n),b(n)) converges on maximum likelihood

Question 23

MCMC provides

Answer 23

a simple metropolis algorithm for generating random samples of points from L(a,b)

Question 24

MCMC

Answer 24

1. sample random initial point P(1) = (a(1),b(1))
2. Centre a new pdf, Q, called the proposal density, on P(1)
3. Sample tentative new point P’=(a’,b’) from Q
4. Compute R = L(a’,b’)/L(a(1),b(1))

see notes for diagram

if R > 1 : P’ is uphill we accept P’
if R < 1 : P’ is downhill we may reject P’

Question 25

whether we accept R < 1

Answer 25

generate a random number x ~ U[0,1] if x < R then accept P' if x > R then reject P'

Question 26

correlation theorem

Answer 26

the FT of the first time domain function, multiplied by the complex conjugate of the FT of the second time domain function is equal to the FT of their correlation

Question 27

the broader the gaussian in the time domain

Answer 27

the narrower the gaussian in the frequency domain

Question 28

sketch the probability density function of the CDF

Answer 28

should be a normal distribution with two probability density functions

Question 29

how a CDF of a random variable can be used to generate a random sample

Answer 29

to generate a sample for p(x), sample y from U[0,1], a unitary number in range 0 -> 1 compute x = P^-1(y) or y = P(x) Then x~p(x) see notes for graph

Question 30

how is a maximum likelihood constructed

Answer 30

1. first consider the best model which fits the data 2. a visual inspection can be used to see if its a uniform, normal... 3. then calculate the likelihood for the chosen distribution 4. the individual likelihoods are then multiplied and a minimisation is used to 'maximise' the likelihood.

Question 31

quantisation noise likelihood function

Answer 31

uniform distirubtion

Question 32

(x(i) - µ)

Answer 32

is the residual used in the least squares problem

Question 33

histogram of sampled values

Answer 33

will show that all values in the interval are equally likely to occur and the histogram should be flat

Question 34

correlations between neighbouring pseudo-random numbers

Answer 34

plotting x(i) versus x(i+1) the data should be randomly scattered and show no pattern

Question 35

autocorrelation

Answer 35

the autocorrelation should be unity for zero lag, and zero for all other values

Question 36

chi-squared test

Answer 36

a confidence limit of p>0.05 will show whether the hypothesis is believable

Question 37

statistics that describe noisy data

Answer 37

noise comes from random distributions i.e. uniform or normal distribution

Question 38

the upper frequency is given by

Answer 38

the Nyquist-Shannon sampling theorem

Question 39

the lower frequency is given by

Answer 39

the lower bound is set by the total data length

Question 40

low pass filter

Answer 40

from Nyquist-Shannon sampling theorem a low pass filter will generate a sinc function in the time domain when converting to the frequency domain, thus will just become a top-hat function this is the representation of our ideal filter.

Question 41

low pass filter sketch

Answer 41

see notes

Question 42

why is a low pass filter important

Answer 42

to reduce the noise to remove the problem of aliasing

Question 43

sketch of no correlation

Answer 43

see notes

Question 44

sketch of positive correlation

Answer 44

see notes

Question 45

sketch of negative correlation

Answer 45

see notes

Question 46

central limit theorem

Answer 46

for any pdf with finite variance σ^2 , as M -> ∞ µ(hat) follows a normal pdf with mean µ and variance σ^2 / M

Question 47

probability density function sketch

Answer 47

see notes

Question 48

means for poisson

Answer 48

number of photons/second counted by a CCD number of galaxies/degree^2

Question 49

if correlation coefficient = 0

Answer 49

then x and y are independent

Question 50

the residuals are equally likely

Answer 50

to be positive or negative and all have equal variance

Question 51

weighted least squares

Answer 51

makes good use of small data sets

Question 52

ordinary least and weighted lest squares plot

Answer 52

see notes

Question 53

chi 2 used when

Answer 53

we know there are definite outcomes no errors on measurement

Question 54

reduced chi 2 used when

Answer 54

we know there is uncertainty or variance in a measured quantity errors on measurement

Question 55

reduced chi 2 degrees of freedom

Answer 55

are the number of data points