Model Fitting

Question 1

{xi…xm}=

Answer 1

random sample from pdf p(x) with mean μ and variance σ^2

Question 2

sample mean

Answer 2

μ hat = 1/m sum from i=1 to M of xi

Question 3

as sample size increases, sample mean

Answer 3

increasingly concentrated near to true mean

Question 4

var(μ hat)=

Answer 4

σ^2/M

Question 5

for any pdf with finite variance σ^2, as M approaches infinity, μ hat follows

Answer 5

a normal pdf with mean μ and variance σ^2/M

Question 6

the central limit theorem exaplains

Answer 6

importance of normal pdf in statistics

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

Question 7

bivariate normal pdf

Answer 7

p(x,y) which is specified by μx, μy, σx, σy, p

often used in the physical sciences to model the joint pdf of two random variables

Question 8

the first four parameters of the bivariate normal pdf are

Answer 8

equal to the following expectation values

E(x)=μx
E(y)=μy
var(x)=σx^2
var(y)=σy^2

Question 9

the parameter p is known as the

Answer 9

correlation coefficient

Question 10

what does the correlation coefficient satisfy?

Answer 10

E[(x-μx)(y-μy)]=pσxσy

Question 11

if p=0, then

Answer 11

x and y are independent

Question 12

what is E[(x-μx)(y-μy)]=pσxσy also known as

Answer 12

the covariance of x and y and is often denoted cov(x,y)

Question 13

what does the covariance define?

Answer 13

how a parameter (x) varies with another parameter (y)

Question 14

p>0

Answer 14

positive correlation
y tends to increase as x increases

Question 15

p<0

Answer 15

negative correlation
y tends to decrease as x increases

Question 16

contours become narrower and steeper as

Answer 16

|p| approaches 1

Question 17

what is pearson’s product moment correlation coefficient

Answer 17

r

given sampled data, used to estimate the correlation between variables

Question 18

if p(x,y) is bivariate normal, then r is

Answer 18

an estimator of p

Question 19

the correlation coefficient is a unitless version of

Answer 19

the covariance

Question 20

if x and y are independent variables, cov(x,y)=

Answer 20

0

so p(x,y)=p(x)p(y)

Question 21

the method of least squares

Answer 21

workhorse method for fitting lines and curves to data in the physical sciences

useful demonstration of underlying statistical principles

Question 22

ordinary least squares

Answer 22

scatter plot of (x,y) is assumed to arise from errors in only one of the two variables

Question 23

ordinary least squares - can write

Answer 23

yi=a+bxi+Ɛi

Question 24

what is Ɛi

Answer 24

the residual of the ith data point

i.e the difference between the observed value of yi and the value predicted by the best fit, characterised by parameters a and b

Question 25

we assume that the Ɛi are

Answer 25

an independently and identically distributed random sample from some underlying probability distribution function with mean zero and variance σ^2 (residuals are equally likely to be positive or negative and all have equal variance)

Question 26

ds/da=0 when

Answer 26

a=a_LS

Question 27

Weighted least squares is an efficient method that makes good use of

Answer 27

small data sets

Question 28

weighted least squares - in the case where σi^2 is constant for all i, the formulae

Answer 28

reduce to those for the unweighted case

Question 29

principle of maximum likelihood is a method to

Answer 29

estimate the parameters of a distribution which fit to observed data

Question 30

principle of maximum likelihood - first

Answer 30

decide which model we think best describes the process of generating the data.

Question 31

Maximum likelihood estimation is a method that will find the values

Answer 31

of mu and sigma that result in the curve that best fits the data

Question 32

Assuming all events are independent, then the total probability of observing all of data is

Answer 32

the product of observing each data point individually (i.e. the product of the individual probabilities)

Question 33

when is chi2 used?

Answer 33

when we know there are definite outcomes e.g. flipping a coin, measure whether email arrival rate is constant in time => no errors on measurement

Question 34

when is reduced chi2 used?

Answer 34

when we know there is uncertainty or variance in a measured quantity e.g. measure flux from a galaxy => errors on measurement

Question 35

poisson distribution, k=

Answer 35

1 (mean)

Question 36

normal distrubution, k=

Answer 36

2 (mean and variance)

Question 37

degrees of freedom=

Answer 37

N-K-1

Question 38

For the reduced Chi2, don’t know number of outcomes, so degrees of freedom are

Answer 38

the number of data points

Question 39

p-value

Answer 39

If the null hypothesis were true, how probable is it that we would measure as large, or larger, a value of chi2 ?

Question 40

standard value to reject a hypothesis

Answer 40

a p-value <0.05

Question 41

If we obtain a very small P-value (e.g. a few percent?) we can interpret this as

Answer 41

providing little support for the null hypothesis, which we may then choose to reject.