Basic Statistical Toolbox

Question 1

mathematical statistics

Answer 1

concerned with theoretical foundations (probability theory)

Question 2

applied statistics

Answer 2

concerned with modelling of data and the errors in our observations to make inferences about the physical system we are observing

Question 3

statistical error

Answer 3

Uncertainty in the measurement of a physical quantity that is essentially unpredictable – just
as likely to yield a measurement that is too large as one that is too small.

Question 4

‘common sense principle’

Answer 4

if we repeat our measurements many times and average the results then average length = ‘true’ length

Question 5

systematic error

Answer 5

Uncertainty in the measurement of a physical quantity that is always systematically too large
or too small. (Measurement is biased)

Question 6

Note that here the systematic error enters not when we make our measurements, but

Answer 6

when we analyse them

Question 7

Systematic flaws in our data analysis methods, rather than in
our data themselves, are just as serious but

Answer 7

easier to fix

Question 8

plausible reasoning

Answer 8

probability measures our degree of belief that something is true

Question 9

prob(X)=1

Answer 9

certain that X is true

Question 10

prob(X)=0

Answer 10

certain that X is false

Question 11

In astronomy we generally measure continuous variables which can
take on

Answer 11

infinitely many possible values

Question 12

with infinitely many possible values, p(X) is no longer a probability but a

Answer 12

probability density function

Question 13

probabilities are never

Answer 13

negative
p(x)>or=0 for all x

Question 14

we compute probabilities by

Answer 14

measuring the area under the pdf curve

ie integral between a and b of p(x) dx

Question 15

normalisation

Answer 15

integral between -infinity and infinity =1

Question 16

important pdfs:

Answer 16

1. poisson
2. uniform
3. central/normal/gaussian

Question 17

examples of poisson pdfs

Answer 17

number of photons counted by CCD
number of galaxies counted by galaxy survey
photons in laser beam

Question 18

poisson pdf assumes

Answer 18

detections are independent and there is a constant rate μ

Question 19

uniform pdf

Answer 19

p(x)=1/b-a when a<x<b
o otherwise

Question 20

what is a cdf

Answer 20

cumulative distribution function

Question 21

cdf P(a)=

Answer 21

integral from -infinity to a of p(x)dx = prob(x<a)

Question 22

the nth moment of a pdf is defined as (discrete case)

Answer 22

sum from x=a to x= of x^np(x)delta x

Question 23

nth moment of a pdf for the continuous case

Answer 23

integral from a to b of x^np(x)dx

Question 24

1st moment is called the

Answer 24

mean or expectation value

Question 25

second moment is called

Answer 25

the mean square

Question 26

the variance for the discrete case is defined as

Answer 26

sum between a and b of p(x)(x-)^2delta x

Question 27

the variance for the continuous case

Answer 27

integral between a and b of p(x)(x-)^2dx

Question 28

the variance is often written as

Answer 28

σ^2

Question 29

σ=sqrt(σ^2) is called

Answer 29

the standard deviation

Question 30

in general, var[x]=

Answer 30

-^2

Question 31

the median divides the CDF into

Answer 31

two equal halves

Question 32

prob(x

Answer 32

prob(x>xmed)=0.5

Question 33

the mode is the values of x for which

Answer 33

the pdf is a maximum

Question 34

for a normal pdf, mean=

Answer 34

median=mode=μ

Question 35

Poisson mean and variance

Answer 35

mean = μ variance = μ

Question 36

uniform mean and variance

Answer 36

mean = 1/2(a+b) variance = 1/12(b-a)^2

Question 37

normal mean and variance

Answer 37

mean = μ variance =σ^2

Question 38

probability =

Answer 38

long run relative frequency of an event given infinite many experimental trials

Question 39

{xi,...,xm} =

Answer 39

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

Question 40

σ^2 and μ are

Answer 40

fixed but unknown parameters

Question 41

sample mean

Answer 41

μ hat = 1/M Σ from i=1 to M of xi

Question 42

E(μ hat) = μ when

Answer 42

taking all samples in distribution

Question 43

as sample size increases, sample mean

Answer 43

increasingly concentrated near to true mean

Question 44

the central limit theorem

Answer 44

for any pdf with finite variance, as M approaches infinity, u hat follows a normal pdf with mean u and variance σ^2/M

Question 45

the central limit theorem explains importance of normal pdf in statistics but still based on

Answer 45

asymptotic behavior of an infinite ensemble of samples that we didn't actually observe

Question 46

variable probability density function example

Answer 46

chi 2